Kepler: Strena seu de Nive Sexangula

Cum non sim nescius, quam tu ames Nihil, non quidem ob pretii vilitatem, sed propter lascivi passeris lusum argutissimum simul et venustissimum: facile mihi est conjicere, tanto tibi gratius et acceptius fore munus, quanto id Nihilo vicinius.

Since I am not unaware how you love Nothing, not indeed on account of the cheapness of its price, but because of the lascivious sparrow’s play, most witty and most charming at once: it is easy for me to conjecture that a gift will be the more pleasing and acceptable to you, the nearer it is to Nothing.

Quicquid id est, quod aliqua Nihili cogitatione tibi allubescat, id et parum et parvum et vilissimum et minime durabile, hoc est pene nihil esse oportet. Qualia cum in rerum natura multa sint, est tamen inter ea delectus. Cogitabis fortasse de uno ex atomis Epicuri: verum id Nihil est.

Whatever it is that, by some cogitation of Nothing, makes you take a liking, that must be both too little and small and most vile and least durable, that is, to be almost nothing. Although many such things exist in the nature of things, yet there is among them a selection. You will perhaps think of one of Epicurus’s atoms: but in truth that is Nothing.

Nihil vero a me habes antea. Eamus itaque per elementa, hoc est per ea, quae sunt in unaquaque re minima.

Nothing indeed you have from me before. Let us therefore go through the elements, that is, through those things which are in each thing minimal.

Primum de Terra, hoc est de Archimedis mei thesauris ne somnies, qui Terram in arenas resolvit, qui pulvisculorum dena millia possidet in uno grano papaveris. Unum enim si subtraxero numerorum illi, myriadum rationes plane confudero. Adde quod hujusmodi corpusculorum figura nec oculis videtur, nec ab Archimede proditur.

First about Earth, that is, do not dream of the treasures of my Archimedes, who resolves the Earth into sands, who possesses ten thousands of tiny dust-motes in a single grain of poppy. For if I subtract one unit from his numbers, I shall plainly have confounded the reckonings of myriads. Add that the figure of little bodies of this kind is neither seen by the eyes, nor put forth by Archimedes.

Nullum igitur in iis ingenium, nulla rei non visae cupido. Est et res durabilis, pulvis, ut quae trabibus vetustate subactis, carie confectis insidens dominatur. Nimium igitur dedero, hoc si dedero.

Therefore no ingenuity in them, no desire for a thing not seen. It too is a durable thing, dust, as that which, sitting upon beams subdued by age, consumed by rot, holds dominion. I shall therefore have given too much, if I grant this.

Ignis porro scintillae, etsi parvae et evanidae, nunquam tamen sunt minores arenulis pyritarum, quae conflictu deteruntur, aut strigmentis prunarum, quae jam inter pulvisculos rejeci. Figurales itaque pyramidas, quas nunquam vidi, Platoni relinquo, ut ex iis arbitratu suo concinnet ignem. Veniendum ad elementa intermedia.

Further, the sparks of fire, although small and evanescent, are nevertheless never smaller than the little grains of pyrites, which are worn down by collision, or than the scrapings of coals, which I have already rejected among the little dust-particles. The figural pyramids, which I have never seen, I leave to Plato, that from them at his own discretion he may assemble fire. We must come to the intermediate elements.

Ventum et fumos dare possem, sed hi venduntur, neque hoc tantum in utribus Islandicis, sed et in chartis, quin et in verbis, idque passim per orbem terrarum. Res itaque pretiosa fumus, et quae magno mihi constat. Neque haec apta ingenio, quia rudis et informis.

I could give wind and smokes, but these are sold, and not only in Icelandic wineskins, but also on papers, indeed even in words, and that everywhere through the orb of lands. Smoke, therefore, is a precious thing, and one which costs me greatly. Nor are these things apt to genius, because rude and formless.

Ad aquas igitur devolvimur. Haerentem in urna guttam sacrosancti vates pro re contemtissima reputant. Et Germani nostri nil minus habent illa vini guttula, quae post cyathum exhaustum super unguem exeutitur ibique haerens mole sua stat.

Therefore we are rolled down to the waters. The sacrosanct seers reckon a drop clinging in the urn as a most contemptible thing. And our Germans count no less that tiny droplet of wine which, after the cup has been drained, is flicked upon the fingernail, and there, clinging, stands by its own weight.

Si hanc obtulero guttam, minus sane dedero, quam ille Persa, vola manus Choaspem Regi suo affundens; honestius etiam munus, vini gutta ex ungue Germani, quam derosum ramentum de ungue inferendentis et vel tantillum negantis Itali. Denique figura guttae globosa jam speculationem pollicetur geometricam: sed vereor, ne et hoc tibi sit nimium, qui tantopere delectaris Nihilo.

If I shall have proffered this drop, I shall indeed have given less than that Persian, who, with the palm of his hand, pours the Choaspes to his king; and the drop of wine from a German’s nail is even a more honorable gift than the scraped scrapings from the nail of an Italian who proffers and yet denies even the least little bit. Finally, the globose figure of a drop already promises a geometric speculation: but I fear lest even this be too much for you, you who so greatly take delight in Nothing.

Quid si ad animalia fiat transitio? Vereor ut noctuas Athenas. Nuper enim apud te vidi volumina rerum singularium et rararum ejus, qui ex veteri Parmenidis schola motum tollit, quia motus partem unam (stilicet praeteritam) perfectam non habet.

What if the transition be made to animals? I fear I would be bringing owls to Athens. For recently at your place I saw volumes of singular and rare matters by the one who, from the ancient school of Parmenides, abolishes motion, because motion does not have one part (to wit, the past) perfect.

Quo in opere cum insint monstra pleraque, haud reor defutura animalcula prodigiosa exilitate. Quanquam nihil opus uti conjecturis. Habes animadversiones Scaligeri in Cardani subtilitates.

In that work, since for the most part there are monsters, I do not think there will be lacking animalcules prodigious for exility. Although there is no need to employ conjectures. You have Scaliger’s animadversions upon Cardan’s Subtleties.

Invenies ibi animal minimum (Exercitatione CXCIV. num. 7.), cuniculum subcutaneum.

You will find there a very small animal
(in Exercise 194, no. 7), a subcutaneous burrower.

Est vero et hoc nimium. Nam cum incedat id animalculum, anima non caret. Igitur animam tibi cur offeram, cui etiam inanimem guttam dare refugio?

Truly, even this
is too much. For since that animalcule proceeds, it does not lack a soul. Therefore why should I offer you
a soul, to whom I even refuse to give an inanimate drop?

Nisi forte exsecto grassatricis bestiae cadavere nova aliqua deprehendi posse speras: de quo viderit D. Jessenius anatomicus.

Unless perhaps, with the cadaver of the marauding beast cut open you hope that some new things can be detected: about which let Dr. Jessenius, the anatomist, see.

Talia dum meditans anxie pontem transeo, confusus super incivilitate mea, qui coram te sine strena comparuissem, nisi quod eadem perpetuo chorda oberrans identidem Nihil affero, nec invenirem, quidnam esset Nihilo proximum, quod ingenii pateretur acumen, commodum accidit, ut vaporibus vi frigoris in nivem coeuntibus flocculi sparsim in vestem meam deciderent, omnes sexanguli, villosis radiis. Eia me Hercule rem quavis gutta minorem, figuratam tamen, eia strenam exoptatissimam Nihil amanti, et dignam quam det mathemeticus, Nihil habens, Nihil accipiens, quia et de coelo descendit et stellarum gerit similitudinem.

While anxiously pondering such things as I cross the bridge, confounded over my incivility, I who would have appeared before you without a strena, except that, harping perpetually on the same string, again and again I bring Nothing, nor could I find what might be next to Nothing, which would endure the acumen of wit, it conveniently happened that, as vapors by the force of cold were coalescing into snow, little flakes fell here and there upon my garment, all six-angled, with shaggy rays. Ha, by Hercules, a thing smaller than any drop, yet figured; ha, a strena most ardently desired by one who loves Nothing, and worthy to be given by a mathematician, Having Nothing, Receiving Nothing, because it both descends from heaven and bears the likeness of the stars.

Redeatur ad patronum, dum durat strenula, ne corporis halitu tepido solvatur in nihilum.

Let it be returned to the patron, while the little New‑Year’s‑gift endures, lest by the tepid breath of the body it be dissolved into nothing.

Atque en fatale nomen. O rem Wackherio gratissimam, Nihil amanti. Nam si Germano quaeras, nix quid sit, respondebit Nihil, siquidem Latine possit.

And lo, the fatal name. O thing most gratifying to Wackherius, a lover of Nothing. For if you ask a German what nix is, he will answer Nothing, provided he can in Latin.

Accipe igitur hanc Nihil accessionem sereno vultu, et si sapis, animam contine, ne denuo nihil accipias.

Receive therefore this Nothing accession with a serene countenance, and if you are wise, hold your breath, lest you receive nothing again.

Dicendum enim est Socrati de saltu pulicis, hoc est quare nives primo casu, priusquam implicentur in majores floccos, perpetuo cadant sexangulae, villosis, ut pennulae, senis radiis.

For one must speak to Socrates about the leap of the flea, that is, why snows at the first fall, before they are enfolded into greater flakes, perpetually fall sexangular, villous, like little feathers, with six rays.

Imo facessat hinc popularis contemtus inscitiaeque leno Aristophanes, quid enim mihi opus Socrate, ipsius fabulae materia? Ipse in regium Psalten respicio, qui inter Dei laudes commemorat, quod det nivem sicut lanam, qua voce, nisi fallor, expressit villosos illos nivulae meae radios. Verisimile enim est, cum sederet fessus, aut staret innixus pedo ad custodiam gregis, vidisse et notasse stellulas hasce nivales, in lanas ovium defluentes ibique adhaerentes.

Nay rather, let vulgar scorn be gone from here, and Aristophanes, the pander of ignorance; for what need have I of Socrates, the very subject-matter of his comedy? I myself look to the royal Psalmist, who among the praises of God records that He gives snow like wool, by which phrase, unless I am mistaken, he expressed those woolly rays of my little snowflake. For it is plausible that, when he sat weary, or stood leaning on his shepherd’s staff for the guarding of the flock, he saw and noted these little snow-stars, flowing down upon the sheep’s wool and there clinging.

Sed ad rem veniamus joco misso. Cum perpetuum hoc sit, quoties ningere incipit, ut prima illa nivis elementa figuram prae se ferant asterisci sexanguli, causam certam esse necesse est. Nam si casu fit, cur non aeque quinquangula cadunt, aut septangula, cur semper sexangula, siquidem nondum confusa et glomerata multitudine varioque impulsu, sed sparsa et distincta?

But let us come to the matter, jest set aside. Since this is perpetual, whenever it begins to snow,
that those first elements of snow bear before them the figure of a six-angled asterisk,
a definite cause must exist. For if it happens by chance, why do they not equally
fall five-angled, or seven-angled, why always six-angled, seeing that they are not yet
confused and conglomerated by a multitude and by a various impulse, but scattered and distinct?

Qua de re cum esset mihi sermo cum quodam nuper, primum inter nos convenit, causam non in materia quaerendam, sed in agente. Materia enim nivis est vapor, is dum oritur ex terra, calore quodam suo subvectus, non alius quam continuus et quasi fluidus est: non igitur distinctus in singulares hujusmodi stellulas.

On which matter, when I had a conversation with a certain man recently, first it was agreed between us, that the cause should be sought not in the matter, but in the agent. For the matter of snow is vapor, this, while it arises from the earth, borne aloft by a certain heat of its own, is nothing other than continuous and as it were fluid: therefore not distinct into individual little stars of this sort.

Quaeras, unde hoc sciam, cum, si talis etiam sit vapor, id oculis cerni non possit, quia vapor pellucidus? Respondeo: vapor existit ex resolutione humoris subterranei, quod arguitur ex ejus levitate et ascensu. In resolutione vero figurae non habent locum.

You may ask whence I know this, since, even if such be the vapor, it can not be seen by the eyes, because vapor is pellucid? I reply: vapor exists from the resolution of subterranean humor, which is evinced by its lightness and ascent. In a resolution, indeed, figures have no place.

Id enim habet figuram ex se, quod se ipso terminatur, cum termini figuram constituant; vapor, resolutione facta, ex generibus humidorum est et fluit, hoc est se ipso non terminatur, nullam igitur figuram retinet, donec condensetur in nivem vel guttam.

For that has a figure from itself which is bounded by itself, since boundaries constitute a figure; vapor, once dissolution has taken place, is of the genera of humors and flows—that is, it is not bounded by itself—therefore it retains no figure, until it is condensed into snow or a drop.

Cum igitur constaret, causam inditae figurae sexangulae esse penes agentem, dubitatum porro fuit, quodnam id esset agens et quomodo ageret, num ut forma insita, an ut efficiens extrinsecum, num ex necessitate materiae efficeret figuram sexangulam, an ex sua natura, puta cui congenitus sit vel archetypus pulchritudinis, quae est in sexangulo, vel finis notitia, ad quem ista forma conducat?

Since, therefore, it was established that the cause of the imparted six-angled figure lay with the agent, there was further a doubt what that agent was and how it acted, whether as an inherent form, or as an extrinsic efficient, whether it effected the six-angled figure from the necessity of the matter, or from its own nature, for instance, as one to whom there is an inborn or archetypal form of the beauty which is in the hexagon, or by a knowledge of the end to which that form conduces?

Ut pateat harum quaestionum discrimen, utamur exemplis nobilibus, sed geometrice descriptis. Nam ad quaestionem nostram plurimum faciet excursus iste.

That the distinction of these questions may be evident, let us employ noble examples, but described geometrically. For this excursus will contribute very greatly to our inquiry.

Si ex geometris quaeras, quo ordine structi sint apum alveoli, respondebunt, ordine sexangulo. Simplex est responsio ex intuitu simplici foraminum seu portarum laterumque, quibus efformantur alveoli. Circumstant n. alveos singulos sex alii, singulis lateribus de intermedio singuli communicantes.

If you ask the geometers in what order the bees’ alveoli (cells) are structured, they will answer: in a hexagonal order. The answer is simple, from a simple intuition/inspection of the openings or “portals” and the sides by which the cells are formed. Indeed, around each single cell there are six others, one apiece communicating from the middle along each of its sides.

At ubi fundos alveorum fueris contemplatus, singulos trinis planis in obtusum descendere videbis angulum. Fundum hunc (carinam potius nuncupes) cum senis alveoli lateribus copulant sex alii anguli, tres altiores trilateri planeque similes imo carinae angulo, tres humiliores quadrilateri interjecti. Praeterea considerandum est, geminum esse alveolorum ordinem, portis aversis in contraria, posticis inter se contiguis et stipatis, angulis carinarum singularum ordinis unius inter angulos tres trium carinarum ordinis alterius insertis, ea arte, ut alveus quilibet non tantum sex lateribus communicet cum cum senis alveis in eodem ordine circumstantibus, sed etiam trinis in fundo planis cum tribus aliis alveis ex ordine averso.

But when you have contemplated the bottoms of the cells, you will see each one descend by three planes to an obtuse angle. You might rather call this bottom the keel; this keel is coupled with the six sides of the cell by six other angles: three higher, trilateral and plainly similar to the angle at the lowest part of the keel, and three lower, quadrilateral, interposed. Moreover, it must be considered that the order of the cells is twin, with the doors turned away to opposite sides, their backs contiguous with one another and packed close, the angles of the keels of the individual cells of the one order inserted among the three angles of the three keels of the other order, by such art that any cell not only communicates by its six sides with the six cells standing around in the same order, but also by three planes at the bottom with three other cells from the opposite order.

Ita fit ut apes singulae novem habeant vicinas, a qualibet vno communi pariete distinctae. Plana carinarum trina omnia inter se similia sunt, ejus figurae, quam geometrae rhombum appellant. Quibus ego rhombis admonitus coepi in geometria inquirere, num quod corpus simile regularibus quinque et Archimedeis quatuordecim ex rhombis meris constitui possit: invenique duo, quorum alterum cognatum sit cubo et octaedro, reliquum dodecaedro et icosaedro (nam cubus ipse tertii vicem sustinet, cognatus duobus tetraedris invicem coaptatis), primum duodecim rhombis clauditur, alterum triginta.

Thus it comes about that each individual cell has nine neighbors, each from each distinguished by a single common wall.
The three planes of the keels are all similar to one another, of that figure which the geometers call a rhombus.
Prompted by these rhombi I began to inquire in geometry whether some body akin to the five regular [solids] and the fourteen Archimedean [solids] could be constituted from pure rhombi:
and I found two, of which one is kindred to the cube and octahedron, the remaining to the dodecahedron and icosahedron (for the cube itself sustains the place of a third, being kindred to two tetrahedra fitted to one another), the former is enclosed by twelve rhombi, the latter by thirty.

Sed primo haec est communis proprietas cum cubo, quod ut anguli octo cuborum octonorum circa idem punctum coaptati locum omnem explent, nullo relicto vacuo, sic rhombici primi obtusi seu trilateri anguli quaterni idem praestent et quadrilateri anguli seni similiter. Itaque strui potest locus solidus ex meris hisce rhombis, sic ut semper quatuor trilateri, ut et sex quadrilateri anguli ad unum et unum punctum concurrant. Et ut summa quaedam fiat: quando locus solidus per cubos aequales ordine recto impletur, tunc unum cubum contingunt alii 32 angulis singulis et praeterea sex quaternis, itaque contingentium sunt octo et triginta.

But first, this is a common property with the cube, that just as the eight angles of eight cubes coapted around the same point fill the whole place, no void being left, so the obtuse, or three-sided, angles of the first rhombic, four at a time, accomplish the same, and likewise the four-sided angles, six at a time. Therefore a solid place can be constructed out of these mere rhombi, so that always four three-sided and likewise six four-sided angles concur to one and the same point. And to make a sort of sum: when a solid place is filled, in straight order, by equal cubes, then one cube is touched by 32 others at single angles and, besides, by six at four apiece; therefore those touching are 38.

At quando impletur locus solidus per rhombica aequalia, tunc unum rhombicum contingunt alia 6 angulis singulis quadrilateris, et praeterea duodecim, angulis quaternis, itaque contingentium quomodocunque sunt octodecim.

But when the solid space is filled by equal rhombics, then one rhombic is touched by another 6 at single quadrilateral angles, and besides by twelve, at four angles each; thus, those touching it, in whatever way, are eighteen.

Haec igitur illa figura geometrica est, regularis quam proxime, impletrix loci solidi, ut sexangulum, quadrangulum, triangulum, consummatores loci plani: haec inquam est, quam apes effingunt in suis alvearibus, nisi hoc tantum demto, quod alveoli carent tectis carinae similibus.

Therefore this is that geometric figure, as near as possible to regular, a filler of solid space, as the hexagon, quadrangle, triangle are consummators of plane space: this, I say, is the one which the bees fashion in their hives, only with this deducted, that the little cells lack roofs similar to a keel.

Si enim etiam haec adderent et quaelibet apis intra alias duodecim seu octodecim abderetur, non pateret ipsi exitus, conclusae circumcirca. Itaque cum tectis non indiguerint, nihil obstitit quo minus latera sena pro modulo corpusculi sui producerent ultra modulum rhomborum in carinis, efficerentque ea illorum altrinsecus dissimilia.

For if they were to add these also, and any bee were hidden within twelve or eighteen others, an exit would not be open to it, being shut in all around. Therefore, since they did not need roofs, nothing stood in the way to prevent them from extending the six sides, according to the modulus of their little body, beyond the modulus of the rhombi in the keels, and from making these unlike those on the other side.

Porro si quis grandius aliquod malum granatum aperiat, videbit acinos plerosque in eandem figuram expressos, nisi quantum impedit series radicum, per quas alimentum illis suum suppeditatur.

Moreover, if anyone opens some rather larger pomegranate, he will see the kernels for the most part impressed into the same figure, except insofar as the series of roots, through which their own nourishment is supplied to them, hinders.

Quaeritur jam in his duobus exemplis, quis sit auctor figurae rhombicae in alveolo apum inque granis mali punici? Materia in causa non est. Nuspiam enim inveniunt apes hujusmodi foliola rhombica in praeparato, quae colligant apes atque coaptent ad effigiendas suas domunculas.

It is asked now in these two examples, who is the author of the rhombic figure in the honeycomb of bees and in the grains of the pomegranate? The material is not the cause. Nowhere indeed do the bees find rhombic leaflets of this sort in a prearranged supply, which they might collect and fit together for the fashioning of their little homes.

Neque verisimile est, in solis malis punicis sponte excrescere acinos in angulos, cum in omnibus aliis fructibus rotundi evadant, qua non inpediuntur, humore suggesto lentos cortices explente et referciente, ut turgescant et qua datur protuberent.

Nor is it likely that, in pomegranates alone, the seeds should of themselves grow out into angles, since in all other fruits they turn out round, where they are not impeded, with the supplied moisture filling and re-stuffing the pliant rinds, so that they become turgid and, where it is permitted, protuberate.

Est igitur in acino quidem punici mali figurae causa in anima plantae, quae pomi procurat incrementum. Sed non est haec adaequata figurae causa, neque enim hoc praestat fructui ex formali proprietate, sed adjuvatur necessitate materiali. Nam cum acini inter initia, dum parvi sunt, rotundi sint, quamdiu spatium ipsis intra corticem sufficit, tandem indurescente cortice, crescentibus vero continue acinis, fit eorum constipacio et compressio, ut et pisorum intra suos oblongos calices.

Therefore in the berry (acinus) of the pomegranate the cause of the figure is in the soul of the plant, which procures the growth of the fruit. But this is not an adequate cause of the figure; for it does not furnish this to the fruit from a formal property, but is aided by material necessity. For since the acini at the beginning, while they are small, are round, so long as space within the rind suffices for them, finally, as the rind hardens, while the acini, however, keep on growing, there arises a crowding and compression of them, as with peas within their oblong calyces.

Sed pisa non habent quorsum cedant: oblongis enim siliquis ex ordine sunt inserta, comprimuntur igitur a duobus tantum lateribus. Acini vero rotundi in malis punicis liberius spatium a principio nacti, facile sese singuli intra ternos ordine adverso protuberantes insinuant, rotunditate sua adjuti, humoremque inde, unde urgentur, declinantes in spatia vacua. Quodsi quis aliquam vim globulorum rotundorum interque sese aequalium ex materia molli constantium concludat in rotundo vase illudque circulis aereis incipiat coarctare undique a plagis omnibus: globuli plurimi exprimentur in schema rhombicum, praesertim si prius illos globulos succussione vasis diligenti locum angustiorem libero rotatu capere permiseris.

Sed peas do not have anywhere to yield: for they are set in order within oblong pods, therefore they are compressed from only two sides. But the round berries in pomegranates, having from the beginning obtained freer space, each easily insinuate themselves among three, bulging in opposite order, aided by their roundness, and deflecting the moisture from the place where they are pressed into the empty spaces. But if someone should confine a certain quantity of round globules, equal among themselves and consisting of a soft material, in a round vessel, and begin to constrict it with bronze rings on all sides from every quarter: very many globules will be pressed out into a rhombic schema, especially if beforehand you have allowed those globules, by careful shaking of the vessel, to take the narrower place by free rotation.

Nam directa globulorum dispositione, quae turbari non possit, compressione facta, cubos etiam efficies.

For with a regular disposition of the globules, which cannot be perturbed, once compression is applied, you will even produce cubes.

In universum enim duobus modis inter se ordinantur globuli aequales in vase aliquo collecti, pro duobus modis ordinationis eorum in aliqua planitie.

In general, equal globules collected in some vessel are arranged among themselves in two ways, according to the two modes of their ordering on some plane.

Nam si errantes in eodem plano horizontali globulos aequales coegeris in angustum, ut se mutuo contingant, aut triangulari forma coeunt, aut quadrangulari; ibi sex unum circumstant, hic quatuor; utrinque eadem est ratio contactus per omnes globulos, demtis extremis. Quinquanguli forma nequit retineri aequalitas, sexangulum resolvitur in triangula: ut ita dicti duo ordines soli sint.

For if, gathering equal globules wandering on the same horizontal plane into a narrow space, so that they touch one another, they come together either in a triangular form or a quadrangular; there six surround one, here four; in both cases the same ratio of contact obtains through all the globules, the extremes excepted. In a pentagonal form equality cannot be retained; the hexagon is resolved into triangles: so that only the two orders just mentioned exist.

Fig. 1

Figure 1

Jam si ad structuram solidorum quam potest fieri arctissimam progrediaris ordinesque ordinibus superponas, in plano prius coaptatos, aut ii erunt quadrati (A) aut trigonici (B); si quadrati, aut singuli globi ordinis superioris singulis superstabunt ordinis inferioris, aut contra, singuli ordinis superioris sedebunt inter quaternos ordinis inferioris. Priori modo tangitur quilibet globus a quatuor circumstantibus in eodem plano, ab uno supra se et ab uno infra se, et sic in universum a sex aliis, eritque ordo cubicus, et compressione facta fient cubi; sed non erit arctissima coaptatio. Posteriori modo praeterquam quod quilibet globus a quatuor circumstantibus in eodem plano tangitur, etiam a quatuor infra se et a quatuor supra se, et sic in universum a duodecim tangetur, fientque compressione ex globosis rhombica.

Now if you proceed to the structure of solids as close-packed as can be made and superpose ranks upon ranks, first coapted in a plane, they will be either square (A) or trigonal (B); if square, either the individual globes of the upper rank will stand upon the individual ones of the lower rank, or conversely, the individual ones of the upper rank will sit among sets of 4 of the lower rank. In the former way, each globe is touched by 4 surrounding it in the same plane, by 1 above itself and by 1 beneath itself, and thus in all by 6 others; and the arrangement will be cubic, and when compression is applied they will become cubes; but it will not be the closest coaptation. In the latter way, besides the fact that each globe is touched by 4 surrounding it in the same plane, it is also by 4 beneath itself and by 4 above itself, and thus in all it will be touched by 12; and under compression they will become, from globose forms, rhombic ones.

Ordo hic magis assimilabitur octaedro et pyramidi. Coaptatio fiet arctissima, ut nullo praeterea ordine plures globuli in idem vas compingi queant. Rursum si ordines in plano structi fuerint trigonici, tunc in caoptatione solida aut singuli globi ordinis superioris superstant singulis inferioris, coaptatione rursum laxa, aut singili superioris sedent inter ternos inferioris.

This ordering will more closely resemble the octahedron and the pyramid. The coaptation will be the tightest, such that in no other arrangement besides this can more globes be compacted into the same vessel. Again, if the orders set in the plane are triangular, then in the solid coaptation either the single globes of the upper order stand upon the singles of the lower—the coaptation being again loose—or the singles of the upper sit among three of the lower.

Priori modo tangitur quilibet globus a sex circumstantibus in eodem plano, ab uno supra et ab uno infra se, et sic in universum ab octo aliis. Ordo assmilabitur prismati, et compressione facta fient pro globulis columnae senum laterum quadrangulorum duarumque basium sexangulorum. Posteriori modo fiet idem, quod prius posteriori modo in quadrilateris.

In the former way each globe is touched by six surrounding in the same plane, by one above and by one below itself, and thus in all by eight others. The order will be assimilated to a prism, and, compression having been made, in place of the little globes there will be columns with six quadrangular sides and two hexagonal bases. In the latter way the same will happen as before in the latter way in the quadrilateral case.

Esto enim B copula trium globorum. Ei superpone A unum pro apice; esto et alia copula senum globorum, C, et alia denum D, et alia quindenum E. Impone semper angustiorem latiori, ut fiat figura pyramidis. Etsi igitur per hanc impositionem singuli superiores sederunt inter trinos inferiores, tamen jam versa figura, ut non apex sed integrum latus pyramidis sit loco superiori, quoties unum globulum degluberis e summis, infra stabunt quatuor ordine quadrato.

Let B be a coupling of three globules. To it
superimpose A, one for the apex; and let there also be another coupling of six globules, C, and another
of ten, D, and another of fifteen, E. Always impose the narrower upon the broader, so that a
figure of a pyramid may be made. Although therefore by this imposition each of the upper ones sat among three of the lower,
nevertheless, the figure now turned over, so that not the apex but the entire side of the pyramid is in the upper position, whenever you strip off one globule
from the topmost, below there will stand four in a square order.

Et rursum tangetur unus globus ut prius a duodecim aliis, a sex nempe circumstantibus in eodem plano, tribus supra et tribus infra. Ita in solida coaptatione arctissima non potest esse ordo triangularis sine quadrangulari, nec vicissim. Patet igitur, acinos punici mali, materiali necessiate concurrente cum rationibus incrementi acinorum, exprimi in figuram rhombici corporis, cum non infestis frontibus pertinaciter nitantur rotundi ex adverso acini, sed cedant expulsi in spatia inter ternos vel quaternos oppositos interjecta.

And again one globe will be touched, as before, by twelve others, namely by six surrounding in the same plane, three above and three below. Thus, in the most tight solid coaptation, there cannot be a triangular order without a quadrangular, nor conversely. It is clear, therefore, that the grains of the pomegranate, material necessity concurring with the ratios of the growth of the grains, are pressed into the figure of a rhombic body, since the round grains do not stubbornly strive with hostile fronts against the opposite grains, but, driven back, yield into the spaces interposed between three or four opposed.

Fig. 2

In alvearibus vero apum ratio est alia. Non n. conglobantur apes confuse, ut acini in malo, sed arbitrariam struunt aciem, omnes capitibus prominentes in unam vel adversam plagam, omnes alvorum extremis invicem obnitentes. Quodsi ex conglobatione hujusmodi existeret figura haec, oporteret alveos apibus superindui ex consistentia exsudati lentoris, ut cochleis contortis solent supercrescere domunculae.

In beehives indeed the method of bees is different. For they do not conglomerate confusedly,
as pips in an apple, but they build a chosen array, all with their heads projecting into
one or the opposite quarter, all mutually bracing with the ends of their bellies. But if from
a conglomeration of this sort this figure were to arise, it would be necessary that the cells be
superinduced upon the bees from the consistency of an exuded viscosity, as little houses are wont
to overgrow coiled snails.

At certum est, apes ipsas suos sibi fingere alveos totamque a fundamentis contignationem exstruere.

But it is certain that the bees themselves fashion their own hives for themselves, and erect the entire framework from the foundations.

Quare ipsa apis natura hunc instinctum habet ex proprietate sua, ut hac potissimum figura aedificet; hic illi archetypus a creatore impressus est, nihil hic materia neque cerae neque corpusculi apis, nihil incrementa possunt.

Wherefore the bee’s very nature has this instinct from its own property, to build especially in this figure; here the archetype has been impressed upon it by the Creator; here neither the matter nor the wax nor the bee’s small body, nothing of increments, can avail.

Hoc animadverso quaeritur jam porro et de fine, non quem apis ipsa consectetur discursu suo, sed quem Deus ipse, apiculae creator, propositum habuit, cum illi has architecturae suae leges praescripsisset.

This having been observed, inquiry is now made further also about the end—not the one which the bee itself pursues in its own course, but the one which God himself, the creator of the little bee, had as his purpose, when he prescribed to it these laws of its architecture.

Atque hic jam demum rursum ingreditur finis destinationem, consideratio corporum materiaeque, tria enim de hoc fine dici possunt. Primum vulgare est apud physicos, qui ad solam quidem sexangularem structuram respiciunt, ut illa cum hiatibus extrinsecus sese repraesentat. Cum enim locum plantum impleant excluso vacuo tantum hae figurae, triangulum, quadrangulum, sexangulum, ex iis sexangulum capacissima est figura.

And here now at last again the end enters upon its destination, the consideration of bodies and of matter; for three things can be said about this end. First is the common view among the physicists, who look only to the hexagonal structure, as it presents itself with openings outward. For when they fill a plane place, with the void excluded, only these figures do so: the triangle, the quadrangle, the hexagon; of these, the hexagon is the most capacious figure.

Capacitatem autem sibi parant apes ad mella condenda.

However, the bees prepare capacity for themselves for storing honey.

Potesque ampliari haec ratio etiam ad solidi considerationem in hunc modum, quod cum solidum spatium non dividatur sine hiatu, nisi in cubos et rhombica, rhombica sunt cubis capaciora. Sed non sufficit haec ratio, nam si capacitatem quaerunt, cur non quaelibet sibi rotundum fingit nidum, quid opus est minutias loci consectari, quasi nullum in toto alveari restet spatium? Probabilior esset haec altera causa, quamvis nec illa sufficiens ob rationes dictas, quod mollia apicularum corpuscula commodius locantur in nido figurae plurium et obtusorum angulorum quaeque cognatior est sphaericae, quam in cubo, qui paucos et longe procurrentes habet angulos, fundum planum, a corpore tereti abhorrentem.

And this rationale can also be extended to the consideration of the solid in this way, that since a solid space is not divided without a gap, except into cubes and rhombics, rhombics are more capacious than cubes. But this rationale does not suffice, for if they seek capacity, why does not each one fashion for itself a round nest; what need is there to chase the minutiae of the place, as if no space remained in the whole hive? More probable would be this other cause, although neither is that sufficient for the reasons stated, namely that the soft little bodies of the bees are more conveniently placed in a nest of a figure with more and obtuse angles—and one more cognate to the spherical—than in a cube, which has few and far‑projecting angles, a flat bottom, disagreeing with a rounded body.

Igitur tertiam causam necesse est addere, qua minuitur ipsis labor, si semper duae communem struant parietem et quod in rectitudine coassationum major firmitudo, ad cratem integram sustinendam, quam si singulae domunculae teretes ideoque compressu faciles fuissent; denique figurae rotundae hiant, cum maxime conjunctae sunt, itaque frigus se per hiatus insinuaret. Quibus omnibus providetur, quod consortia tecta urbis habent, ut Vergilius canit.

Therefore it is necessary to add a third cause, by which the labor is diminished for them, if always two build a common wall; and that in the straightness of the joinings there is greater firmness for sustaining the intact lattice, than if the individual little houses were rounded and thus easy to compress; finally, round figures gape when they are most closely joined, and so cold would insinuate itself through the gaps. All of which is provided for by what the communal roofs of a city possess, as Vergil sings.

Has igitur rationes materialem necessitatem respicientes ita puto sufficere, ut hoc loco non existimem philosophandum de perfectione et pulchritudine vel nobilitate figurae rhombicae, neque satagendum, ut essentia animulae quae est in ape, ex contemplatione figurae, quam fabrieatur, eliciatur, quale quid nobis fuisset inceptandum, si usus figurae nullus apparuisset.

Therefore I think these reasons, looking to material necessity, suffice, so that in this place I do not think one should philosophize about the perfection and beauty or nobility of the rhombic figure, nor be bustling that the essence of the little soul that is in the bee be elicited from contemplation of the figure which it fabricates, a sort of undertaking we would have had to initiate, if no use of the figure had appeared.

Idem de malo punico intelligendum. Apparet necessitas materialis, quae acinos perducit ad rhombicum, succedentibus incrementis. Itaque vanum est, de essentia animae in hac arbore cogitare, quae rhombicum potissimum efficiat.

The same is to be understood about the pomegranate. A material necessity is apparent, which brings the seeds to a rhombic shape, as successive increments ensue. Therefore it is vain to think about the essence of the soul in this tree as that which would chiefly produce the rhombic form.

Contra si quaeratur, cur omnes adeo arbores et frutices (aut certe pleraeque) florem explicent forma quinquanguli, numero scilicet foliorum quinario, quem florem in pomis et pyris sequitur fructus dispositio, in eodem vel cognato numero, quinario vel denario, quini intus loculi continendis semenibus, dena filamenta, quod et obtinet in cucumeribus et id genus aliis, hic inquam locum habet speculatio pulchritudinis aut proprietatis figurae, quae animam harum plantarum characterisavit. Et detegam obiter cogitationes meas super hac re.

On the other hand, if it be asked why all trees and shrubs to such a degree (or at least the greater part) unfold the flower in the form of a pentagon, namely with the number of leaves being five, and why in apples and pears the disposition of the fruit follows that flower in the same or a cognate number, five or ten—the inner locules for containing seeds being five, the filaments ten—which also obtains in cucumbers and others of that kind, here, I say, there is room for a speculation on the beauty or the property of the figure which has characterized the soul of these plants. And I will incidentally disclose my thoughts upon this matter.

Duo sunt corpora regularia, dodecaedron et icosaedron, quorum illud quinquangulis figuratur expresse, hoc triangulis quidem, sed in quinquanguli formam coaptatis. Utriusque horum corporum ipsiusque adeo quinquanguli structura perfici non potest sine proportione illa, quam hodierni geometrae divinam appelant. Est autem sic comparata, ut duo minores proportionis continuae termini juncti constituant tertium, semperque additi duo proximi constituant immediate sequentem, eadem semper durante proportione, in infinitum usque.

There are two regular bodies, the dodecahedron and the icosahedron, of which the former is explicitly figured by pentagons, the latter indeed by triangles, but fitted together in the form of a pentagon. The structure of each of these bodies, and indeed of the pentagon itself, cannot be perfected without that proportion which modern geometers call the divine. It is so constituted that the two lesser terms of a continued proportion, when joined, constitute the third, and that always the two nearest, when added, constitute the immediately following term, the same proportion ever continuing, on to infinity.

In numeris exemplum perfectum dare est impossibile. Quo longius tamen progredimur ab unitate, hoc fit exemplum perfectius. Sint minimi 1 et 1, quos imaginaberis inaequales; adde fient 2, cui adde majorem 1, fient 3, cui adde 2, fient 5, cui adde 3, fient 8, cui adde 5, fient 13, cui adde 8, fient 21. Semper enim, ut 5 ad 8, sic 8 ad 13, fere, et ut 8 ad 13, sic 13 ad 21 fere.

To give a perfect example in numbers is impossible. The farther, however, we proceed from unity, the more perfect the example becomes. Let the least be 1 and 1, which you will imagine as unequal; add them, they become 2; to which add the greater 1, they become 3; to which add 2, they become 5; to which add 3, they become 8; to which add 5, they become 13; to which add 8, they become 21. For always, as 5 is to 8, so is 8 to 13, nearly; and as 8 is to 13, so is 13 to 21, nearly.

Ad hujus proportionis se ipsam propagantis similitudinem puto effictam esse facultatem seminariam, itaque in flore praefertur seminariae facultatis ãvçóùí vexillum quinquangulum. Mitto cetera, quae ad hujus rei confirmationem jucundissima contemplatione possent adduci. Sed proprius illis debetur locus.

After the similitude of this proportion that propagates itself, I think the seminary (seed-bearing) faculty has been fashioned; and thus in the flower there is borne the pentagonal banner of the seminary faculty in bloom. I pass over the rest, which could be adduced for the confirmation of this matter by a most pleasant contemplation. But a more proper place is owed to them.

Nunc haec exempli tantum causa praemisimus, ut in rimanda figura nivis sexangula simus instructiores exercitatioresque.

Now we have premised these things for the sake of example only, so that, in scrutinizing the sexangular figure of snow we may be better instructed and more exercised.

Cum enim proposuissemus inquirere originem figurae hujus in nive inter causas extrinsecas et intrinsecas, inter externas primum sese offererbat frigus. Condensatio sane est a frigore, per condensationem vero vapor coit in figuram stellae, videbatur igitur frigus illi figuram praestare stellae. Tunc itum est ad considerationem aliam, an frigus sit natura quaedam, ut medicorum calor?

Since we had proposed to inquire the origin of this figure in snow among the extrinsic and intrinsic causes, among the external first presented itself cold. Condensation, to be sure, is from cold, and through condensation the vapor comes together into the figure of a star; therefore cold seemed to bestow upon it the figure of a star. Then it was gone to another consideration, whether cold is a certain nature, like the physicians’ heat?

Videbatur n. esse mera privatio, cui neque mens, sexanguli fabricatrix, nec omnino operatio ulla propria.

It seemed, namely, to be a mere privation, to which belonged neither mind, the fabricatrix of the hexagon, nor any proper operation at all.

Sed ne misceamus quaestiones, maneat frigori condensatio, potuit condensatio fieri, ut videtur, in formam globosam rectius. Imo si consideretur frigus late fusum et vapor illi superficietenus occurrens, magis est consentaneum, ut condensatio fiat in formam omnino planam, superficiei similem, et eam quorumcunque terminorum. Ut si tota vaporis extima superficies ex frigore densitatem, ex densitate pondus, ex pondere casum, ex casu comminutionem in frustula seu bracteas nancisceretur, utique non omnes bracteae, quin imo paucissimae, ac nescio an ullae evadent sexangulae, praesertim radiis adeo concinne striatis.

Yet, lest we mix questions, let condensation remain to cold; condensation could, as it seems, more properly have been made into a globose form. Nay rather, if one consider cold widely diffused and the vapor meeting it only surfacewise, it is more consonant that the condensation be made into a form entirely plane, like a superficies, and with whatever boundaries. As if the whole outer surface of the vapor, from the cold, were to acquire density, from density weight, from weight a fall, from the fall a comminution into little fragments or thin plates (bracteae), assuredly not all the plates—nay rather very few, and I know not whether any at all—would turn out hexagonal, especially with rays so neatly striated.

Admonebant istae striae rei illius, quae contingit in hypocaustis vapidis, brumali rigore pertusas fenestras obsidente. Luctantur circa illas angustias frigidus aer et vapor. Quoties enim sese mutuo contingunt, calor superiora petit, frigus inferiora.

Those striae reminded one of that phenomenon which happens in warm hypocausts, with brumal rigor besieging perforated windows. Around those constrictions cold air and vapor wrestle. For as often as they touch one another, heat seeks the upper parts, cold the lower.

Est enim in calido dilatatio materiae, in frigido densitas et pondus pellitque calida sursum. Vapore igitur confertim exire nitente, fit fuga vacui, ut et frigidus aer confertim irrumpat, unde limbi patentis fenestrae vel rimulae frigidissimi efficiuntur. Ad eos limbos quicquid appellit vaporis, continuo gelatur succeditque in illam materiam frigus aeque magnum, ut, quicquid porro vaporis ad hanc appellit pruinam, et ipsum geletur, appositione continua, intercedente tamen seseque introrsum insinuate rectis lineis aere frigido, qua alternatione ingressus et egressus illae pruinosae vaporis consistentiae strias sortiuntur et acutos radios.

For in the warm there is dilatation of matter; in the cold, density and weight; and the warm drives upward. Therefore, as the vapor strives to go out in a mass, there arises a flight from the void, so that cold air likewise rushes in in a mass; whence the edges of an open window or little crack are rendered most frigid. Whatever portion of vapor touches those edges is immediately frozen; and into that material there succeeds a cold equally great, so that whatever further vapor touches this hoarfrost is itself frozen, by continuous apposition, the cold air, however, intervening and insinuating itself inward along straight lines; by which alternation of ingress and egress those frosty consistencies of vapor obtain striae and sharp rays.

Nihil ad hoc instar de figuratione nivulae nostrae dici potest. Nam quinam hic ingressus, qui exitus, quae angustiae, quae lucta in patentissimis aeris campis? Concessero, inter cadendum ex alto per vapidum aerem fieri aliquam ad villos appositionem a contingentibus vaporibus.

Nothing of this sort can be said about the figuration of our little cloud. For what entrance here, what exit, what narrownesses, what struggle, in the most open fields of the air? I will concede that, while falling from on high through vapid air, some apposition to the hairs is made by the contacting vapors.

At quare sex locis, quodnam senarii principium? quis capitellum, ante quam caderet, in sex effigiavit cornua frigida? Quae causa statuens in illa superficie jam jam condensanda sex puncta, ad quae seni circum radii connectantur?

But why in six places, what is the principle of the senary?
Who fashioned the little head, before it fell, into six frigid horns?
What cause, appointing on that surface just now about to be condensed, six
points, to which six rays are connected around?

Cum itaque causa externa, frigus, haec efficere nequeat, internam aliquam esse necesse est vaporisque vel comitem vel alio quocumque modo propriam.

Since therefore an external cause, cold, cannot effect these things, it is necessary that there be some internal one, of vapor either a companion or, in whatever other way, proper.

At haec perpendentem subiit admiratio, cur radii non potius in omnem ambitum sphaericum disponantur? Cur si internus calor est hujus rei auctor, in plana tantummodo superficie operatur, qui undiquaque aequaliter se habet, non vero in sola plana superficie vaporis inest?

But as I was weighing these things, a wonder arose: why are the rays not rather disposed through every spherical ambit? Why, if an internal heat is the author of this matter, does it operate on a plane surface only, which is alike on every side, and does the vapor not inhere on a plane surface alone?

Dum in his luctor meditationibus, dum ratio postulat radios in omnem ambitum distributos, incidit, quod alias saepe cum admiratione spectavi, stellulas hujusmodi non primo statim casus momento sterni super planitiem, sed particulis nonnullis sublimes teneri, denique temporis mora subsidere in planitiem. Ex illa ratiocinatione, veluti patre, et ex hac experientia, veluti matre, nata est mihi opinio ista, stellulas istas cadendum trinis constare villosis diametris, decussatim ad unum punctum coaptatis , sex extremitatibus in orbem aequaliter distributis, ita ut tribus tantum villosis radiis incidant, reliquis trinis (cui sunt incidentium oppositi in iisdem rectis diametris), in sublimi stent, donec flexis iis, quibus stellula sustinebatur, reliqui hactenus sublimes in eandem planitiem cum prioribus intermediis locis defluant.

While I wrestle in these meditations, while reason demands rays distributed into every ambit, it occurred—the thing which at other times I have often looked upon with admiration—that little stars of this kind are not laid upon the plane at the very first moment of the fall, but are held aloft by some particles, and finally, after a delay of time, settle onto the plane. From that ratiocination, as it were the father, and from this experience, as it were the mother, there was born to me this opinion: that those little stars, as regards their falling, consist of three shaggy diameters, fitted crosswise (decussatim) to one point , with six extremities equally distributed in a circle, so that they touch with only three shaggy rays, the remaining three (which are opposite the points of incidence on the same straight diameters) standing aloft, until, those by which the little star was supported being bent, the rest hitherto aloft flow down onto the same plane with the former, in intermediate places.

Hujus opinionis vim prosequar per totum, postea demum an vera sit examinabo, ne fortassis importuna vanitatis detectio me prohibeat, quod institui, verba de re Nihili facere.

I will prosecute the force of this opinion throughout, afterwards at last I will examine whether it is true, lest perhaps an inopportune detection of vanity prohibit me from what I have instituted, to make words about the matter of Nothing.

Hoc igitur in causa positum esto, quaecunque causa sit horum sex radiorum, eam undique aequaliter fusam esse in omnes plagas, ut si frigus est causa senum radiorum, frigus igitur singulas vaporis portiunculas circumstare aequaliter, aut aequalibus certe intervallis undique, sin calor internus, et hunc in omnes sphaerae plagas ex uno et eodem centro operari.

Let this, therefore, be posited in the case: whatever the cause of these six rays may be, that it is everywhere equally diffused into all regions; so that, if cold is the cause of the six rays, then cold surrounds the individual little portions of vapor equally, or at least at equal intervals on all sides; but if it is internal heat, then this likewise operates into all the regions of the sphere from one and the same center.

At sic nondum discussa, sed translata est quaestio. Nam nondum patet, quare non quinque vel septem, sed omnino sex villosi radii coaptentur ex eodem centro?

But thus the question is not yet discussed, but transferred. For it is not yet evident why not five or seven, but altogether six hairy rays are fitted together from the same center?

Et si quaeras a geometris, quanam in figura tres diametri sese orthogonaliter seu in forma crusis duplicis in eodem centro secent, is respondebit, in octaedro, connexis angulis oppositis. Octaedron enim habet sex angulos. Quare igitur fit, ut nix inter cadendum, priusquam complanetur, tribus villosis diametris se invicem orthogonaliter secantibus imitetur ipsum ôï óêåëåôïí octaedri, ut si radiorum extrema vicina rectis duodenis connectas, integrum octaedri corpus repraesentassis?

And if you ask the geometers in what figure three diameters cut one another orthogonally, that is, in the form of a double cross, at the same center, he will answer: in the octahedron, with opposite angles connected. For the octahedron has six angles. Why then does it happen that snow, while falling, before it is flattened, with three villous diameters cutting one another orthogonally, imitates the very skeleton of the octahedron, so that if you connect the neighboring extremities of the rays with twelve straight lines, you would have represented the entire body of the octahedron?

Quae causa igitur, quod in hos tres villosos radios potius fit condensatio, quam in globum integrum?

What cause, then, that condensation rather occurs into these three villous rays, than into an entire globe?

Possum quidem dicere modum unum, quo ista fiant materiali necessitate, sed is assumit aliquid, quod rursum plus habet admirationis, quam hoc ipsum, quod jam erat demonstrandum. Dicam tamen, si forte ex comparatione multorum falsorum eliciatur veritas. Esto ut vapor, quando primum frigus irruens sentiscit, coaguletur in sphaerulas certae quantitatis.

I can indeed say one mode by which these things happen by material necessity, but it assumes something which in turn has more of admiration than this very thing which already was to be demonstrated. Nevertheless I will say it, if perchance from a comparison of many false things the truth may be elicited. Let it be supposed that vapor, when it first senses the onrushing cold, is coagulated into spherules of a certain quantity.

Hoc est consentaneum. Nam ut in aqua gutta minimum naturale est de fluido, propterea, quod aqua suo pondere non diffluit amplius, quando est infra guttae quantitatem, sic etiam facile concedi potest, inesse in vaporis materia tenacitatem aliquam, qua possit resistere frigori, in certa aliqua quantitate, puta guttae vapidae.

This is consonant. For just as in water the drop is the natural minimum of the fluid, because water by its own weight does not diffuse any further when it is below the quantity of a drop, so too it can easily be conceded that in the matter of vapor there inheres a certain tenacity, by which it can resist the cold, within some determinate quantity—say, that of a vapor-drop.

Fig. 3

Secundo esto, ut hae sphaerae vapiddae se invicem contingant in certa dispositione, puta quadranguli in plano, cubica in solido, qua de supra, sic enim tangetur sphaerula quaelibet ab aliis sex, quarum solae quatuor hic in plano depingi possunt, quinta et sexta intelligenda est superponi et supponi. His ita positis et assumtis, frigore vero per spatia irruente, sphaerulae a contactu uno ad oppositum erunt munitae contra frigus, itaque versus centra quidem sphaerularum fiet condensatio, sic tamen, ut etiam versus diametros contactuum, quibus scilicet locis tutae sunt a frigore.

Secondly, let it be that these vaporous spheres touch one another in a certain disposition, say quadrangular in the plane, cubic in the solid, as above; for thus any spherule will be touched by six others, of which only four can be depicted here in the plane, the fifth and the sixth being to be understood as superposed and subposed. With these things thus set and assumed, and the cold indeed rushing in through the spaces, the spherules will be fortified against the cold from one contact to the opposite; and so there will be condensation toward the centers of the spherules, yet in such a way that also toward the diameters of the contacts, namely in those places where they are safe from the cold.

Verum non immerito, ut praedixi, quaerat aliquis, qua vi sic disponantur sphaerulae in directum?

But not without reason, as I predicted, someone may ask: by what force are the little spheres thus disposed in a straight line?

Si materialiter fieri aliter non posset, jam peractum esset negotium. At possunt materialiter duobus aliis modis disponi, ut supra dictum. At praeterea possunt omnes tres ordines ordinati confundi, ut fiat dispositio varia.

If materially it could not be done otherwise, the business would already have been completed. But they can materially be arranged in two other ways, as said above. But furthermore all three ordered orders can be confounded, so that a varied disposition may be made.

An hanc adsciscemus dispositionis hujus causem, quod in hac sola dispositio est sibi ipsi undique similis et puncta contactuum distribuuntur aequaliter, in ceteris nequaquam? Etsi enim, ut supra dictum, globi singuli a duodenis aliis tanguntur, at spatia inter globos alternis triangula et quadrangula sunt, hic omnia undique quadrangula. Illic diametri quaedam duo oppositorum contactuum sese secant orthogonaliter, reliqua quatuor non item, hic omnes tres diametri sese secant aequaliter et orthogonaliter.

Shall we adopt this as the cause of this arrangement, that in this alone the arrangement is on all sides similar to itself and the points of contact are distributed equally, by no means so in the others? Although indeed, as said above, the individual globes are touched by twelve others, yet the spaces between the globes are alternately triangular and quadrangular, here they are everywhere quadrangular. There certain two diameters of opposite contacts intersect one another orthogonally, the remaining four not so; here all three diameters intersect equally and orthogonally.

Illic connexis extremis diametrorum fit cuboctoedron, hic octaedron intra sphaerulam quamlibet.

There, with the extremities of the diameters connected, a cuboctahedron is formed, here an octahedron within any spherule.

Praestantia quidem hinc patet dispositionis directae prae obliqua, at causa nondum comparet, quae sphaeras hac potius quam illa ratione disponat. Num facit hoc frigus? At quomodo?

The preeminence indeed of the direct disposition over the oblique is evident from this, but the cause does not yet appear which would dispose the spheres by this method rather than that. Does cold do this? But how?

Nam si quid agit, condensat aut penetrat materiam, qua hiat illa, aut qua debiliter resistit. Et ut largus sim, directam quidem in profundum dispositionem causari possit descensu rectilineo versus terram, at in transversum unde haec directio?

For if it does anything, it either condenses or penetrates the matter, where that gapes, or where it resists weakly. And, to be generous, the direct arrangement into the depth indeed could be caused by a rectilinear descent toward the earth, but in the transverse, whence this direction?

Restat igitur, ut calor internus vaporis hanc guttarum dispositionem cubicam efficiat, si modo est cubica ipsarum dispositio, hoc est si Nihil nostrum est Aliquid.

It remains, therefore, that the internal heat of the vapor produces this cubic arrangement of the drops,
if indeed their arrangement is cubic, that is, if our Nothing
is Something.

Huc autem devoluta re, jam perinde est, sive calor quamlibet guttam se ipso in formam octaedri disponat, sive totam materiam in seriem stellarum ordinatam dispescat atque ita singularum spaerarum internam dispositionem per externam universarum seriem adjuvet. Neutrobique casu ordo existere potest tam constanter, ubi confusio, ut hic quidem, in proclivi est.

But with the matter having come down to this point, it is now all the same, whether heat by itself arranges any droplet whatsoever into the form of an octahedron, or divides the whole material into an ordered series of stars and thus aids the internal disposition of individual spheres by the external series of all. In neither case can order arise by chance so constantly, where confusion, as here indeed, is the easier course.

Sed et argumenta sunt, ut potius credamus, singulas guttas, sine ope experni contactus, se ipsis disponi. Etenim, si figura singularum oriretur ab ordine et contactu mutuo plurium, necesse foret, omnes invicem aequales esse stellulas. Jam vero magnum inter eas cernitur discrimen magnitudinis.

But there are also arguments that we should rather believe the individual drops, without the aid of external contact, to be disposed by themselves. For indeed, if the figure of the individual ones arose from the order and mutual contact of several, it would be necessary that all the little stars be equal to one another. Yet in truth a great difference of magnitude is discerned among them.

Ipsa quin etiam multarum ordinatio multum habet insolentiae.

Nay even the very ordering of many has much of strangeness.

Nihil itaque profecimus, nisi pateat modus, quo calor internus guttam vapidam in tribus diametris, forma octaedrica vel certe sexangula firmet, ut ad cas fiat materiae per condensationem collectio. (Memineris praedictum, de nihilo esse opinionem de trina decussatione trium diametrorum.)

We have therefore accomplished nothing, unless the method be made evident by which the internal heat may stabilize the vapid drop along three diameters, in an octahedral or at least hexangular form, so that by condensation there may be a gathering of matter to these. (Remember what was said above, that the opinion about a triple decussation of three diameters is worthless.)

Possit aliquis existimare, volitare villosa ista ramenta solitaria interque cadendum decussatim cocurrere fortuito. Verum id falsum est. Non enim perpetuo trina, non in punctis mediis, non ad unum punctum concurrereat.

Someone might suppose that those villous filaments, solitary, flit about and
in the act of falling run together decussately, fortuitously. But that is false. For they do not perpetually
in threes, nor at the middle points, nor to one single point do they converge.

Adde quod villi omnes a centro seu stellae seu decussationis geminae aversi extrorsum porriguntur, pene ut foliola in ramis abiegnis, quod argumento est, in centro nidulari vim formatricem indeque in omnes plagas aeualiter sese didere.

Add that the hairs all, turned away from the center—whether of a star or of a twin decussation—stretch outward, almost like little leaflets on fir branches, which is an argument that in the center the formative force nests, and from there has disseminated itself equally into all quarters.

Sed fortassis haec causa est trium diametrorum, quod totidem sunt diametri plagarum in animalibus? Habent enim superas, inferas, anteriores, posteriores, dexteras, sinistras partes. Si quis hoc dixerit, meae is opinioni appropinquabit, sed praeter opinionem in paradoxa pertrahetur concessione sua.

But perhaps this is the cause of the three diameters, that there are just as many diameters of the regions in animals? For they have upper, lower, anterior, posterior, right, and left parts. If someone should say this, he will approach my opinion, but beyond expectation he will be drawn into paradoxes by his own concession.

Primum enim consideret, quae natura sit hujus caloris, quae similitudinem animalis architectetur in stellula nivis. Deinde videat, cui bono? Quid enim animali commune cum nive?

First, let him consider what the nature of this heat is, which architects the likeness of an animal in a little snow-star.
Then let him see, to what benefit? What, indeed, does an animal have in common with snow?

Nix ad vitam, qua caret, plagis istis opus non habet. Tertio perpendat, ipsas animalis partes non tam ad figuras geometricas cubumque, primam solidarum figurarum, velut ad archetypum suum accommodatas, quam necessitate quadam ad finem obtinendum directas. Prima enim superi et inferi distinctio est a loco, quae est Terrae superficies, pedes igitur deorsum vergunt, ut contra pondus corporis nitantur, caput sursum est, ut nervos imbre opportuno continue humectet, utque oculi et aures a planitie remotissimi plurimam ejus circumferentiam in conspectu habeant, obstaculis remotis, denique ut cibus pondere, potus humore suo praecipitatus in suum locum descendat, neque continua (ut in plantis uno loco fixis) attractione opus haberet.

Snow, for life, which it lacks, has no need of those zones. Third, let him weigh that the animal’s parts themselves are not so much accommodated to geometric figures and to the cube, the first of solid figures, as to their archetype, as they are directed by a certain necessity to obtain an end. For the first distinction of above and below is from place, which is the surface of the Earth; therefore the feet incline downward, so that they may brace against the weight of the body, the head is upward, so that it may continuously moisten the nerves with a timely “shower,” and so that the eyes and ears, being farthest from the plain, may have the greatest circumference of it in view, obstacles removed; finally, that food by its weight, and drink, precipitated by its own moisture, may descend into its proper place, and would not need continuous attraction (as in plants fixed in one place).

Altera anticae et posticae distinctio tributa est animantibus ad motus exercendos, qui in recta linea super Terrae superficiem tendit a loco ad locum. Itaque duae hae diametri necessario orthogonaliter se mutuo secant signantque superficiem. At cum animalia non possint esse superficies, sed necessario corpora accipiant, tertiam diametrum dextri et sinistri ex ratione corpulentiae necesse fuit accedere, qua fit animal quasi geminum, ut esset etiam in incessu moventis et moti discrimen alternis.

The other distinction of anterior and posterior is assigned to living beings for the exercising of motions, which in a straight line over the Earth's surface tends from place to place. Therefore these two diameters necessarily intersect one another orthogonally and mark the surface. But since animals cannot be surfaces, but must by necessity take on bodies, a third diameter of right and left had to be added by reason of corporeity, whereby the animal becomes as it were twin, so that even in gait there might be a discrimination between the mover and the moved by alternation.

Non igitur, quae cubica sunt, hominis gerunt similitudinem propter aliquam figurae pulchritudinem, sed homo cubi acquisivit similitudinem, quasi concinnatam ex variis usibus seu elementis.

Therefore, things which are cubic do not bear the likeness of man on account of some pulchritude of figure, but man has acquired the likeness of the cube, as if concinnated from various uses or elements.

Itaque omnibus examinatis, quae occurrebant, sic ego sentio, causam figurae in nive sexangulae non aliam esse, quam quae est figurarum in plantis ordinatarum numerorumque constantium. Ac cum in his nihil fiat sine ratione summa, non quidem quae discursu ratiocinationis inveniatur, sed quae primitus in creatoris fuerit consilio et ab eo principio hucusque per mirabilem facultatum animalium naturam conservetur, ne in nive quidem hanc ordinatam figuram temere existere credo.

Therefore, with all the things that presented themselves examined, thus I judge: the cause of the six-angled (hexagonal) figure in snow is no other than that which is of the ordered figures in plants and of constant numbers. And since in these nothing is done without a highest reason, not indeed one that is found by the discourse of ratiocination, but one which from the first was in the creator’s counsel and from that beginning until now is preserved through the marvelous nature of the faculties of living beings, I do not believe that even in snow this ordered figure exists at random.

Est igitur facultas formatrix in corpore Telluris, cujus vehiculum est vapor, ut humana anima, spiritus; adeo ut nullus uspiam existat vapor, quin ut calore quodam id effectus est, quod esse dicitur, puta vapor, eodemque calore conservatur, ut id esse pergat, sic ratione etiam formatrice, quam alii calorem opificem dicunt, contineatur.

There is therefore a formative faculty in the body of Earth, whose vehicle is vapor, as the human soul has the spirit; to such a degree that no vapor exists anywhere, without its having been by a certain heat effected to be that which it is said to be, namely vapor, and by the same heat it is conserved, that it may go on being, thus also by a formative reason, which others call an artificer-heat, it is contained.

Sed duarum objectionum solutione, quod reliquum est de opinione mea, declarabo. Etenim objicere possis: in plantis finem subsequentem, qui est constitutio certi corporis naturalis, arguere, rationem formatricem in aliqua materia praecessisse; ubi enim media ad certum finem ordinata, ibi ordo, ibi nullus casus, ibi mera mens, mera ratio, in nivis vero formatione finem nullum spectari posse, neque fieri per figuram sexangulam, ut nix perduret aut corpus naturale definitum certae et durabilis formae fiat. Respondeo, rationem formatricem non tantum agere propter finem, sed etiam propter ornatum, nec solum tendere ad corpora naturalia efficienda, sed etiam solere ludere in fluxis, quod multis fossilium exemplis patet.

But by the solution of two objections I will declare what remains of my opinion. For indeed you might object: in plants the subsequent end, which is the constitution of a certain natural body, argues that a formative reason has preceded in some matter; for where the means are ordered to a certain end, there is order, there no chance, there pure mind, pure reason; but in the formation of snow no end can be regarded, nor does it come about through the hexagonal figure that snow endures, or that a natural body defined with a certain and durable form is made. I reply that the formative reason acts not only on account of an end, but also for ornament, and not only tends to bring natural bodies into being, but is also wont to play in things in flux, which is evident from many examples of fossils.

Quorum ego universorum rationem a ludicro (dum dicimus naturam ludere) ad hanc seriam intentionem transfero, quod puto, calorem, qui hactenus tutabatur materiam, ubi a circumstanti frigore vincitur, ut hactenus ordine agebat (ratione quippe formatrice imbutus), ordine pugnabat, sic jam suo quodam ordine et fugae sese comparare pedemque reffere et diutius haerere in sparsis istis et ordinatim veluti per aciem distributis ramis, quam in tota reliqua materia, atque sic curae habere, ut (quod de Olympiade referunt historiae *) non inhoneste nec inverecunde cadat.

Of all of which I transfer the account from the ludic (while we say that nature plays) to this serious intention: that
I think the heat, which hitherto was defending the matter, when it is overcome by the surrounding cold, just as up to now it was acting with order (imbued, indeed, with formative reason), was fighting with order; so now, in a certain order of its own, it also prepares itself for flight and withdraws its foot, and
clings longer upon those scattered and, as if along a battle line, orderly distributed branches, than upon all the rest of the matter, and thus takes care that (as the histories report concerning Olympias *)) it may not fall dishonorably nor shamelessly.

Alius aliquis objiciat, plantis singulis singulas esse facultates animales, cum seorsim etiam subsistant corporum plantarum singula, proptereaque nil esse mirum, singulis etiam singulas aptari figuras. In nivis vero qualibet stellula peculiarem fingere animam, per esse ridiculum, quare ne quidem figuras nivis eodem modo ex animae opere ut in plantis deducendas.

Let someone else object that, in plants, to each individual there belong individual animal faculties, since the individual things of the bodies of plants also subsist separately, and therefore there is nothing marvelous that to individuals likewise individual figures be fitted. But in snow, to fashion in any little star a peculiar soul is utterly ridiculous, wherefore not even the figures of snow are to be deduced in the same way from the work of soul as in plants.

Respondeo, rem utrinque similiorem esse, quam, qui haec objicit, credere possit. Demus, plantis singulis singulas esse facultates, at eae omnes soboles sunt unius et ejusdem facultatis universalis, quae in Terra inest quaeque se habet ad plantas, ut facultas aquae ad pisces, facultas humani corporis ad pediculos, canni ad pulices, ovilli ad aliud genus pediculorum. Non enim omnes plantae ex semine, pleraeque Ýî áýôïìáôïõ primum ortae, etsi sese porro seminent.

I reply, the matter is more similar on both sides than the one who objects these things could believe. Let us grant that to single plants there are single faculties, yet all these are the offspring of one and the same universal faculty, which is in the Earth and which bears itself toward plants as the faculty of water to fishes, the faculty of the human body to lice, of the canine to fleas, of the ovine to another kind of lice. For not all plants are from seed; most have at first arisen spontaneously, although thereafter they sow themselves.

Facultas enim Terrae, quae se ipsa una est et eadem, dividit sese in corpora et cum corporibus inque ea inolescit et pro cujusque materiae conditione interna externisve aliud atque aliud architectatur. Ita in vapore quoque, quem totum tota possederat anima, nihil mirum, si frigore divisionem totius continui moliente, ob contractionem partium circa partes ipsa, ut circa singula tota formando occupetur.

For the Faculty of the Earth, which by itself is one and the same, divides itself into bodies and with the bodies grows into them, and according to each matter’s internal or external condition architects now one thing, now another. Thus in vapor too, which the soul as a whole had wholly possessed, there is nothing marvelous if, as cold labors to effect a division of the entire continuum, on account of the contraction of the parts, the soul itself is occupied in forming itself as a whole around the parts, so that around each single one it is engaged in forming a whole.

O vere mortuam vitam sine philosophia. Hanc enim in nive formatricem facultatem si scivisset illa Aesopicae fabellae adultera, persuadere marito potuisset, se ex nive concepisse spurioque suo non tam facile fuisset orbata calliditate mariti.

O life truly dead without philosophy. For this formative faculty in snow
if that adulteress of the Aesopic fable had known, she could have persuaded her husband
that she had conceived from snow, and would not so easily have been deprived of her spurious child
by the callidity of her husband.

Dixi de auctore figurae; restat ut inquiramus de figura ipsa, sive illa existat ex decussatione trium diametrorum, quod hactenus est inter supposita, sive inde ab origine sit sexangula, de quo postea. Nunc pergendum in tramite coepto. Causa igitur, cur haec facultas octaedri dispositionem angulorum potius imitetur, haec esse possit.

I have spoken about the author of the figure; it remains that we inquire about the figure itself, whether it exists from the decussation of three diameters, which up to now is among the suppositions, or whether from the origin it is hexangular, about which later. Now we must proceed on the path begun. Therefore, the cause why this faculty rather imitates the octahedron’s disposition of angles may be this.

Primum universum genus animorum geometricis et regularibus sive cosmopoeticis figuris cognatum est, quod multis documentis probari potest. Cum enim animi sint quaedam exemplaria Dei creatoris, certe in Dei creatoris mente consistit Deo coaeterna figurarum harum veritas. Amplius, cum certissimum sit, ipsos etiam animos penitissima sui essentia recipere quantitates, sine materia physica, an cum ea, non disputo: consentaneum est, figuratas potius recipere quantitates, quam rudes, si figuratas, quare figuras regulares, solidas, quia animi sunt non superficierum, sed corporum solidorum.

First, the whole genus of souls is cognate with geometric and
regular, or cosmopoietic, figures, which can be proved by many documents
and evidences. For since souls are certain exemplars of God the creator, surely in
the mind of God the creator there subsists the truth of these figures coeternal with God. Furthermore,
since it is most certain that souls themselves, in their most inmost essence, receive
quantities, whether without physical matter or with it—I do not dispute—it is congruent that they
receive figured rather than rude quantities; if figured, then regular figures, solid ones, because souls are not of surfaces,
but of solid bodies.

Est autem inter regulares solidas prima, cubus, primogenita, parens ceterarum. Ejus vero femina quasi quaedam est octaedron, habens tot angulos, quot cubus plana eorumque centra, quibus singulis singuli respondent ex octaedro anguli.

Now among the regular solids the first is the cube, the firstborn, the parent of the rest. Its, as it were, female is the octahedron, having as many angles as the cube has planes (faces), and the centers of these, to each single one of which a single angle from the octahedron corresponds.

Portiuncula itaque materiae vapidae deserenda, si figuram debet recipere quod jam fecimus consentaneum, primum cubum arripiet ejusque socium octaedron. Quorsum supra etiam materialis alludebat necessitas, globorum aequalium in unum acervum confusorum. Confundebantur enim et adumbrabantur in punctis contactuum rudimenta cubi octaedrique.

Therefore, a little portion of vapid matter must be set apart, if it ought to receive a figure—which we have already deemed consonant; it will first seize the cube and its associate, the octahedron. To this the material necessity mentioned above was also alluding: of equal globes thrown together into one heap. For at the points of contact the rudiments of the cube and the octahedron were being confounded and adumbrated.

At cur octaedri figuram potius quam cubi? An quia cubus est figura dilatationis, octaedron collectionis? Jam vero et materia et vis calorifica colliguntur impetita hostiliter a frigore.

But why the figure of the octahedron rather than that of the cube? Is it because the cube is the figure of dilatation, the octahedron of collection? And now indeed both matter and the calorific force are gathered together when hostilely assailed by cold.

Unde vero certum sit, illam esse dilatationis figuram, hanc collectionis? Nempe quia octo anguli, quibus illa foris diditur, iidem in hac intus centrum circumsistunt eodem numero. Etenim si cubo adimas angulos suos octonos resectos lateribus aequalibus introrsumque componas, plane constitues octaedron.

Whence, then, is it certain that that is the figure of dilatation, this of collection?
Namely because the eight angles by which that one is spread outward, the same in this stand around the center within in the same number.
For if you take away from the cube its own eight angles, cut off with equal faces, and compose them inward, you will plainly constitute an octahedron.

Et cubus in plures, sc. in octo angulos diffunditur, octaedron in pauciores, puta sex.

And the cube is diffused into more, namely into eight angles, the octahedron into fewer, say six.

Ajunt gemmarii, naturalia in adamantibus inveniri octaedra, perfectissimae et limatissimae formae. Id si est, multum nos confirmat. Nam facultas animalis, quae in Terra indidit adamanti formam octaedri, ex peuitissimo sinu suae naturae depromtam, eadem cum vapore progressa de Terra, figuram eandem indidit et nivi ex vapore illo consistenti.

Gem-cutters say that natural octahedra are found in diamonds, of a most perfect and most polished form. If that is so, it greatly confirms us. For the animate faculty, which in the Earth imparted to the diamond the form of an octahedron, drawn forth from the most inmost bosom of its nature, the same, having advanced with the vapor from the Earth, imparted the same figure also to snow condensing from that vapor.

Quanquam, quod decussationem trium diametrorum attinet, in ea non magis inest octaedri quam cubi forma. Illic anguli, hic centra planorum connectuntur hujusmodi tribus diametris, illic anguli diametrorum ad centrum, hic angulus, qui corpus finit, exprimitur. Frustra igitur de totius figurae electione satagimus, ubi est utriusque rudimentum saltem.

Although, as concerns the decussation of three diameters, in it the form of the octahedron is no more present than that of the cube. There the angles, here the centers of the planes are connected by three diameters of this sort; there the angles of the diameters at the center, here the angle which bounds the body, is expressed. Therefore we strive in vain about the choice of the whole figure, where there is at least a rudiment of each.

Quo vero abripior stultus ego, qui dum pene Nihil donare affecto, pene etiam Nihil ago, quia ex hoc pene Nihilo pene mundum ipsum, in quo omnia, efformavi, cumque ab animula minutissimi animalculi supra refugerim, jam ter maximi animalis, globi Telluris, animam in nivis atomo exhibeo?

But whither am I being swept away, foolish as I am, who, while I strive to bestow almost Nothing, almost also do Nothing, since out of this almost Nothing I have fashioned almost the world itself, in which are all things, and although from the little soul of the most minute little animal I have shrunk back above, now thrice greatest animal, the globe of Earth, I display the soul in an atom of snow?

Itaque pedem referam et sedulo dabo operam, ut quod donavi quodque dixi, id Nihil sit. Fiet autem id, si quam cito nivula mea liquescit, tam cito ratiunculas istas ego contrariis ratiunculis profligavero atque annihilavero.

Therefore I will draw back my foot and will sedulously take pains, so that what I have donated and what I have said may be Nihil. But this will come to pass, if as quickly as my little snow melts, so quickly I shall have routed those little arguments with contrary little arguments and annihilated them.

Dum enim ista scibo, rursum ninxit et confertius quam nuper. Contemplatus sum sedulo corpuscula nivis, cadebant igitur omnia radiosa, sed duorum generum; quaedam minuta valde, radiis circumcirca insitis, incerto numero et simplicibus sine villis, sine striis, erantque subtilissimi; in centro vero colligati ad grandiusculum globulum, atque horum erat maxima pars. Interspargebantur autem secundi generis rariores sexangulae stellulae earumque nulla aliter nisi plana, neque volitabat neque cadebat, villis etiam in eandem planitiem cum caule suo compositis.

While indeed I was writing these things, it snowed again and more densely than lately. I contemplated carefully the little corpuscles of snow; they were all falling radiant, but of two kinds: some very minute, with radii set all around, of uncertain number and simple, without hairs, without striae, and they were most subtle; in the center, however, gathered into a rather larger little globule, and these made up the greatest part. Interspersed, however, were rarer little six-angled stars of the second kind, and none of them otherwise than flat, nor did it flutter nor fall, the little hairs also arranged into the same plane together with its stalk.

Vergebat autem inferius deorsum radiolus septimus, quasi radix aliqua, in quam cadentes incumbebant eaque sustinebantur sublimes aliquandiu; quod me supra non fugit, sed sinistre exceptum est, ac si terni diametri non essent in eodem plano. Itaque non minus quod hactenus dixi, quam quo de dixi, a Nihilo quam proxime abest.

But a seventh little ray was inclining downward beneath, as if some root, upon which the falling ones would lean, and by which they were held aloft for a while; which did not escape me above, but was taken amiss, as if the three diameters were not in the same plane. Therefore, both what I have said thus far and that about which I have spoken come as near as possible to Nothing.

Primum genus grumosum puto esse ex vapore jam pene deserto a calore, et jamjam in guttas aqueas condensando. Itaque et rotunda sunt, et figuram pulchram non sortiuntur, deserta jam ab architecto, et radiosa sunt undique, iis principiis, quae supra ad contemplationem pruinosae consistentiae in fenestris sunt adhibita.

I think the first kind to be grumose, from vapor now almost deserted by heat, and
just now condensing into aqueous drops. Therefore they are both round, and the figure
beautiful they do not obtain, already deserted by the architect, and they are radiant on every side,
by those principles which above were applied to the contemplation of the frosty consistency on
the windows.

In secundo vero genere, quod est stellularum, locum nullum habet contemplatio cubi vel octaedri, neque ullus guttarum contactus, cum plana incidant, non ut supra sum opinatus, decussata trinis diametris.

In the second kind, indeed, which is of little stars, the contemplation of the cube or of the octahedron has no place, nor any contact of drops, since they impinge with planes, not, as I supposed above, decussated with three diameters.

Etsi igitur formatrix anima hic quoque locum suum tuetur manetque in causa, de electione tamen figurae quaestio est redintegranda. Primum cur plana? An quia non recte supra ademi plana formatricibus corporum?

Although therefore the formative soul here also maintains its own place and remains the cause, nevertheless the question about the choice of the figure must be renewed. First, why planes? Or is it because I did not rightly above take planes away from the shapers of bodies?

Nam in omnibus floribus inest quinquangulum planum, non dodecaedrum solidum. Tunc causa planae figurae vere haec esset, quod frigus calidum vaporem in aliqua planitie tangit, nec ita totum vaporem aequaliter circumstat cun stellulae gignuntur, ut cum grumi cadunt.

For in all flowers there is a plane pentagon, not a solid dodecahedron. Then the cause of the plane figure would truly be this: that cold touches the hot vapor on some flat plane, and does not thus surround the whole vapor equally when the little stars are generated, as when clumps fall.

Cur autem sexangula? An quia ex regularibus haec prima est vere plana, et quae in nullum corpus secum colligatur? Nam trigonus, tetragonus, pentagonus corpora efficiunt.

But why hexagonal? Is it because among the regulars this is the first that is truly plane, and which is gathered into no body? For the trigon, tetragon, and pentagon effect bodies.

An quia sexangulum sternit planitiem, excluso vacuo? At idem facit triangulum, quadrangulum. An quia proxima haec circulo ex iis, quae planitiem sternunt, excluso vacuo?

Or because the hexagon paves the plane, the void being excluded? But the triangle and quadrangle do the same. Or because this is the closest to the circle among those which pave the plane, the void being excluded?

An hoc discriminis inter facultatem sterilia figurantem, et alteram illam, quae foecunda figurat, ut illa triangula vel sexangula faciat, haec quinquangula? An denique ipsa hujus formatricis natura in intimo sinu suae essentiae particeps est sexanguli?

Or is this the distinction between the faculty that figures sterile
things, and that other one which figures fecund things: that the former makes triangles or
hexagons, the latter pentagons? Or finally, is the very nature of this formatrix
in the inmost bosom of its essence a participant of the hexagon?

Ex quinque adductis causis prima, secunda et tertia hoc sibi usurpant, facultatem formatricem e re nata consilium capere et pro opportunitate campi aciem instruere, ut quia pugna calidi vaporis et frigidi aeris in planitie existit non per corpulentiam, ipsa quoque figuram eligat, quae planitierum est potius, quam corporum. Itaque et materialis necessitatis rationem haberet in secunda et tertia. Nam prima causa sola sexanguli proprietate freta est, respiciens decentem congruentiam hujus figurae ad hanc pugnam.

Of the five adduced causes, the first, the second, and the third claim this for themselves, the formative faculty, as the matter arises, to take counsel and, according to the opportunity of the field, to draw up the battle-line; since the battle of hot vapor and cold air exists on a plane not by bulk, it likewise chooses a figure which belongs rather to plane-surfaces than to bodies. And thus it would also take account of material necessity in the second and the third. For the first cause relies solely on the property of the hexagon, having regard to the seemly congruence of this figure for this battle.

In plano pugna, necessario igitur figura plana, at non necessario figura talis, quae ad nullum corpus secum ipsa coeat, sed ideo solum talis, quia ut corporibus physicis figurae respondent, quae solidum ambeunt, sic planitiebus figurae, quae solidum non ambeunt, hic decentia formalis spectatur, non necessitas materialis.

On a plane is the battle,
therefore necessarily a planar figure, but not necessarily a figure of such a sort as does not itself cohere with any body, but only for this reason such, because just as to physical bodies there correspond figures which encompass a solid, so to planes figures which do not encompass a solid,
here formal decency is regarded, not material necessity.

An in secunda et tertia hoc dicendum esset, necessitate etiam materiali eligi a formatrice sexangulum, ne quid scilicet relinquatur vacuum, et ut commodius fieri possit collectio vaporis in nivis consistentiam.

Or in the second and third should this be said, that even by material necessity the hexagon is chosen by the formatrix, so that, namely, nothing is left a vacuum, and that the collection of vapor into the consistency of snow may be made more conveniently.

In circulo enim commodissime fieret, at quia circelli vacua spatia relinquunt, ideo circuli similior eligitur figura. Verum huic causae jam supra fuit opposita inaequalitas stellularum, quarum alique minutissimae sunt, radiis etiam exilissimis et simplicibus, sine villis. Quod est argumento, non magnam aliquam vaporis planitiem simul coire in nivem, sed seorsim planitiunculas minimas, alias post alias easque inaequales.

For indeed in a circle it would be made most commodiously; but because little circles leave empty spaces,
therefore a figure more similar to a circle is chosen. But to this cause there was already set above the inequality of the little stars, of which some are very minute, with rays even most slender and simple, without little hairs. Which is an argument that not some great flat plane of vapor coalesces at once into snow, but separately the tiniest little flatlets, one after another, and unequal at that.

Non habet ergo locum consideratio exclusionis vacui, quae regnat tantum in divisione integrae superficiei in sexangula aequalia. Ita fiet, ut secunda et tertia causa e numero deleantur, nisi quatenus ad primam redigi possunt, ut formatrix facultas sexangulum eligat, nulla materiae spatiorumque necessitate coacta, sed solum decentia hac invitata, quod alias sexangulum struat planitiem excluso vacuo sitque (ex iis figuris, quae idem possunt) circuli simillima.

Therefore the consideration of the exclusion of the void has no place, which reigns only in the division of an entire surface into equal hexagons. Thus it will come about that the second and third cause are deleted from the number, except insofar as they can be reduced to the first: that the forming faculty choose the hexagon, compelled by no necessity of matter and of spaces, but invited only by this decency, inasmuch as otherwise the hexagon constructs a plane with the void excluded and is (among those figures which can do the same) most similar to the circle.

Quarta quidem causa sic unda consistere nequit. Nam alba lilia trinis senisque effigiantur foliis, et sterilia non sunt, eodem modo multi calices florum fere silvestrium. Nisi forte hoc discriminis sit, quod fructus sub flore quinquangulo enascitur carnosus, ut in pomis pyrisque, aut pulpaceus, ut in rosa, cucumeribus, seminibus intra carnem vel pulpam abditis.

The fourth cause indeed cannot stand thus. For white lilies are fashioned in threes and in sixes with petals, and they are not sterile; in the same way many calyces of flowers, for the most part wild ones. Unless perhaps this is the distinction: that the fruit beneath the quinquangular flower grows fleshy, as in apples and pears, or pulpaceous, as in the rose, in cucumbers, with the seeds hidden within the flesh or pulp.

At sub flore sexangulo nil enascitur, nisi semen in sicco loculo, estque velut in flore fructus. Aut est hoc forte discrimen, quod nullus flos sexangulus in arboribus et fruticibus, sed in herbis et fere bulbaceis. Vel consideret alius ipsos succos, an aliquod in iis discrimen secundum figuras florum.

But under the hexagonal flower
nothing is produced, except seed in a dry locule, and it is as if in the flower
there is the fruit. Or perhaps this is the distinction: that no hexagonal flower is on trees
and shrubs, but on herbs and almost on bulbaceous plants. Or let another consider the juices themselves,
whether there is any distinction in them according to the figures of the flowers.

Res mihi nondum comperta est, itaque sufficiat leviter admonuisse alios de hac quarta causa.

The matter is not yet ascertained to me, and so let it suffice to have lightly admonished others concerning this fourth cause.

Pro quinta causa faciunt opera hujus formatricis facultatis alia, ut crystalli, omnes sexanguli, cum adamantes octaedrici sint rarissimi. Sed formatrix Telluris facultas non unam amplectitur figuram, gnara totius geometriae et in ea exercita. Vidi enim Drasdae in aede Regia, cui stabulo nomen, exornatum abacum aere argentoso, ex quo quasi efflorescebat dodecaedron avellanae parvae magnitudine, dimidia parte exstans.

In favor of the fifth cause there speak other works of this formative faculty, such as crystals, all hexagonal, whereas octahedral diamonds are very rare. But the formative faculty of Earth does not embrace a single figure, being skilled in all geometry and practiced in it. For I saw at Dresden in the Royal hall, whose name is the Stable, an abacus adorned with silvery bronze, from which, as it were, a dodecahedron was efflorescing, of the size of a small hazelnut, projecting halfway.

Exstat et in descriptione thermarum Bollensium icosaedri pars anterior inter fossilia. Itaque verisimile est, hanc facultatem formatricem pro diverso humore diversam fieri. In vitriolo crebra est figura cubica rhombica, in nitro sua est figura.

There also exists in the description of the Bollensian baths the anterior part of an icosahedron among the fossil specimens. Therefore it is plausible that this formative faculty becomes diverse according to the differing humor. In vitriol the rhombic-cubic figure is frequent; in nitre it has its own figure.

Dicant igitur chymici, an in nive sit aliquid salis, et quodnam salis genus, et quam illud alias induat figuram. Ego namque, pulsatis chymiae foribus, cum videam, quantum restet dicendum, ut causa rei habeatur, malo abs te, Vir solertissime, quid sentias, audire, quam disserendo amplius fatigari.

Let the chymists, then, say whether there is anything of salt in snow, and what genus of salt it is, and what figure it otherwise puts on. For I, having knocked at the doors of chymistry, since I see how much remains to be said in order that the cause of the matter may be ascertained, would rather hear from you, Most Skillful Man, what you think than be wearied further by discoursing.

*) De Polyxena, cum ad sepulchrum Achillis immolaretur, hic apud Euripidem versus est: ðïëëçí ðñïíïéáí åß÷åí åýó÷çìùò ðåóåéí. Eundem accommodat Plinius jun. in epistolis virgini cuidam Vestali, quam Domitianus vivam defodit.

*) About Polyxena, when she was immolated at the sepulcher of Achilles, this verse is in Euripides: she had much forethought to fall decorously. The same line is applied by Pliny the Younger in the Letters to a certain Vestal virgin, whom Domitian buried alive.

FINIS

THE END