Balbus: BALBI EXPOSITIO ET RATIO OMNIUM FORMARUM

[1] Notum est omnibus, Celse, penes te studiorum nostrorum manere summam, ideoque primum sedulitatis meae inpendium iudiciis tuis offerre proposui. Nam cum sibi inter aequales quendam locum deposcat aemulatio, neminem magis conatibus nostris profuturum credidi quam qui inter eos in hac parte plurimum possit. Itaque quo cultior in quorundam notitiam veniat, omnia tibi nota perlaturus ad te primum liber iste festinet, apud te tirocinii rudimenta deponat, tecum conferat quidquid a me inter ipsas armorum exercitationes accipere potuit.

[1] It is known to all, Celsus, that the sum of our studies rests with you, and therefore I resolved first to offer the first fruits of my diligence to your judgments. For since emulation demands for itself a certain place among equals, I thought that no one would be more useful to our efforts than he who among them can most in this matter. Therefore, that this book may come to the acquaintance of more cultivated men, having to transmit all things known to you, let it hasten first to you, let it lay down the rudiments of its apprenticeship with you, and let it compare with you whatever it could receive from me amid the very exercises of arms.

Et si meretur publica conversatione sufferre universorum oculos, a te potissimum incipiat: quod si illi parum diligentem adhibitam curam esse credideris et in aliqua cessasse videbimur parte, non exiguum laboris mei consequar fructum, quod te monente malignorum lucri fecerim existimationem. quaeso itaque, si non est inprobum, habeat apud te quandam excusationem, quod non potuerit eo tempore consummari, quo genus hoc instrumenti ferventibus studiis nostris disparatum est. Omnium enim, ut puto, liberalium studiorum ars ampla materia est; cui in hac modica re nequid deesset, ingenti animo admoveram vires.

And if it deserves, by public circulation, to endure the eyes of all, let it begin above all with you: and if you shall believe that too little diligent care was applied and that in some part we seem to have been remiss, I shall not reap a small fruit of my labour, since by your warning I would have made the reputation of having sought the profit of the malicious. I therefore beg, if it is not improper, that it have with you a certain excuse, namely that it could not be completed at that time at which this kind of work was produced by our fervent studies. For the art of all liberal studies, as I think, is a wide subject; to which, lest anything be lacking in this modest matter, I had applied my powers with a great spirit.

Intervenit clara sacratissimi imperatoris nostri expeditio, quae me ab ipsa scribendi festinatione seduceret. Nam dum armorum magis exerceor cura, totum hoc negotium velut oblitus intermiseram, nec quicquam aliud quam belli gloriam cogitabam. At postquam primum hosticam terram intravimus, statim, Celse, Caesaris nostri opera mensurarum rationem exigere coeperunt.

The famous expedition of our most sacred emperor intervened, which drew me away from the very haste of writing. For while I was more occupied with the exercise of arms, I had altogether, as it were forgetful, let this whole business lapse, and thought of nothing else but the glory of war. But after we first entered hostile territory, immediately, Celse, the operations of our Caesar began to demand an accounting of measurements.

Erant dandi interveniente certo itineris spatio duo rigores ordinati, quibus in tutelam commeandi ingens vallorum adsurgeret molis: hos invento tuo operis decisa ad aciem parte ferramenti usus explicuit. Nam quod ad synopsim pontium pertinet, fluminum latitudines dicere, etiam si hostis infestare voluisset, ex proxima ripa poteramus. Expugnandorum deinde montium altitudines ut sciremus, venerabilis diis ratio monstrabat.

There were, to be laid out at certain intervening stretches of the route, two ridges arranged, by which a vast mass of ramparts would rise for the protection of passage: when your discovery of the work was made, he deployed these, having cut the iron part of the implement to the line toward the battle. For, as to the synopsis of the bridges, to speak of the widths of the rivers — even if the enemy had wished to harass us, we could do so from the nearest bank. And so that we might know the heights of the mountains to be attacked, a venerable scheme showed them to the gods.

Quam ego quasi in omnibus templis adoratam post magnarum rerum experimenta, quibus interveni, religiosius colere coepi, et ad consummandum hunc librum velut ad vota reddenda properavi. Postquam ergo maximus imperator victoria Daciam proxime reseravit, statim ut e septentrionali plaga annua vice transire permisit, ego ad studium meum tamquam ad otium sum reversus, et multa velut scripta foliis et sparsa artis ordini inlaturus recollegi. Foedum enim mihi videbatur, si genera angulorum quot sint interrogatus responderem 'multa': ideoque rerum ad professionem nostram pertinentium, in quantum potui occupatus, species qualitates condiciones modos et numeros excussi.

Which I, as if worshipped in all temples, after the experiences of great affairs in which I took part, began to cultivate more religiously, and, to complete this book as if to discharge vows, hastened. Therefore after our greatest emperor by victory recently opened Dacia, as soon as he permitted the annual change to pass from the northern quarter, I returned to my study as to leisure, and gathered many things, as if written on leaves and scattered, to bring them into the order of the art. For it seemed foul to me, if asked how many kinds of angles there are, to answer “many”; and therefore, as far as I was able, occupied with matters pertaining to our profession, I examined the kinds, qualities, conditions, modes, and numbers.

Per que satis ampla mediocritatis meae opinio servabitur, si illa vir tantae auctoritatis studentibus profutura iudicaveris.

And by these things a sufficiently ample opinion of my mediocrity will be preserved, if you shall have judged that that man of so great authority will be of service to students.

[2] Ergo nequid nos praeterisse videamur, omnium mensurarum appellationes conferamus. [Nam mensura non tantum ista de qua loquimur appellatur, sed et quidquid pondere aut capacitate aut animo finitur mensuram eque quam longitudinem appellant.] Quid ergo mensura sit de qua quaeritur, tractemus.

[2] Therefore, lest we seem to have neglected anything, let us set together the appellations of all measures. (For mensura is not only that thing of which we speak called so, but also whatever is bounded by weight or capacity or by mind — they call this mensura and also length.) What then the measure is of which inquiry is made, let us treat.

[3] Mensura est conplurium et inter se aequalium intervallorum longitudo finita, ut pes per unciam, per pedem decempeda, per decempedam actus, per passum stadium, per stadium miliarium, et his similia.

[3] A measure is the finite length of several intervals equal among themselves, as for example a foot by an inch, a decempeda (ten-foot) by a foot, an actus by the decempeda, a stadium by a passus (pace), a miliarium (mile) by a stadium, and similar such things.

[4] Mensurarum appellationes quibus utimur sunt duodecim, digitus uncia palmus sextans pes cubitus gradus passus decempeda actus stadium miliarium. Minima pars harum mensurarum est digitus: siquid enim infra digitum metiamur, partibus respondemus, ut dimidiam aut tertiam.

[4] The names of the measures which we use are twelve: digitus (digit), uncia (inch), palmus (palm), sextans (sextant), pes (foot), cubitus (cubit), gradus (step), passus (pace), decempeda (ten‑foot), actus (actus), stadium (stade), miliarium (mile). The smallest part of these measures is the digit: for if we measure anything less than a digit, we reckon by parts, as a half or a third.

[5] Vncia habet digitum unum et tertiam partem digiti. Palmus habet digitos IIII, uncias III. Sextans, que eadem dodrans appellatur, habet palmos III, uncias VIIII, digitos XII.

[5] An uncia has one digit and a third part of a digit. A palm has 4 digits, 3 unciae. The sextans, which is likewise called dodrans, has 3 palms, 9 unciae, 12 digits.

Pes habet palmos IIII, uncias XII, digitos XVI. In pede porrecto semipedes duo. In pede constrato semipedes IIII.

The foot has 4 palms, 12 unciae, 16 digits. In an extended foot there are two half-feet. In a contracted foot there are 4 half-feet.

[6] In pede quadrato semipedes VIII. Cubitus habet sesquipedem, sextantes duas, palmos VI, uncias XVIII. Gradus habet pedes duo semis.

[6] In the squared foot there are 8 half-feet. The cubit has a sesquiped (one and a half feet), two sextants, 6 palms, 18 inches. A step has 2½ feet.

Passus habet pedes quinque. Decempeda, quae eadem pertica appellatur, habet pedes X. Actus habet longitudinis ped. CXX, Latitudinis ped.

A passus has five feet. The decempeda, which is called the same pertica, has 10 feet. An actus has a length of 120 feet, a breadth of feet.

CXX. Stadium habet pedes DCXXV, passus CXXV. Miliarium habet passus mille, milia pedum V, stadios VIII.

120. A stadium has 625 feet, 125 paces. A mile has 1,000 paces, 5,000 feet, 8 stadia.

[7] [Mensurae aguntur generibus duodecim. Digitis. Digitus est in pede pars XVI.

[7] [Measurements are reckoned in twelve kinds. Digits. A digit is a one-sixteenth part of the foot.

Vnciis. Vncia est in pede pars XII. Palmis.

On unciæ. An uncia is a twelfth part of the foot. Of palms.

Palmus IIII. Sextantibus. Sextans, quae eadem dodrans appellatur, habet uncias VIIII, digitos XII.

Palm 4. By sextants. The sextans, which is likewise called a dodrans, has 9 unciae, 12 digits.

Pedibus. Pes palmos IIII. Cubitis.

By feet. A foot is 4 palms. By cubits.

Cubitus pedem semis. Gradibus. Gradus habet pedes II. Passibus.

A cubit is half a foot. By steps. A step has 2 feet. By paces.

Passus habet pedes V. Decempedis. Decempeda pedes X. Actibus. Actus habet pedes CXX.

A passus has 5 feet. Decempedis. A decempeda has 10 feet. Actibus. An actus has 120 feet.

Stadiis. Stadium habet pedes DCXXV. Miliariis.

Stadia. A stadium has 625 feet. Miliaries.

Miliarium habet p. V. Pes prostratus sic observabitur. Ducis longitudinem per latitudinem: facit embadon. Pes quadratus sic observabitur.

The miliarium has p. 5. The prostrate foot shall thus be observed. Of the leader, length by breadth: it makes embadon. The squared foot shall thus be observed.

Longitudinem per latitudinem metiemur, deinde per crassitudinem: et sic efficit pedes solidos. Pes quadratus concavus capit amforam trimodiam. In centuria agri iugera CC, modii DC. In circuitu ped.

We shall measure length by breadth, then by thickness: and thus it produces solid feet. A square hollow foot holds a three‑modius amphora. In a centuria of land there are 200 iugera, 600 modii. In circuit in feet.

VIIIIDC habet. In ea pedum IICCCC per IICCCC, passus CCCCLXXX per CCCCLXXX, actus XX per XX, cubita DC per DC. Pedes ut in cubitos redigamus, semper duco octies, et sumo partem XII: erunt cubita. Cubita vero ut in pedes redigamus, semper duco duodecies, et sumo partem octavam: erunt pedes.]

VIIIIDC has. In it feet 2,400 per 2,400, paces 480 per 480, acts 20 per 20, cubits 600 per 600. To reduce feet into cubits I always take eightfold, and take the twelfth part: there will be cubits. But to reduce cubits into feet I always take twelvefold, and take the eighth part: there will be feet.

[8] Mensurae aguntur generibus tribus, per longitudinem et latitudinem et altitudinem. Hoc est rectum planum solidum. rectum est cuius longitudinem sine latitudine metimur, ut lineas, porticus, stadia, miliaria, fluminum longitudines, et his similia.

[8] Measurements are taken in three kinds, by length and breadth and height. This is a rectilinear plane-solid. "Rectum" is that whose length is measured without breadth, as lines, porticoes, stadia, miles, lengths of rivers, and similar things.

Planum est quod Greci epipedon appellant, nos constratos pedes; in quo longitudinem et latitudinem habemus; per quae metimur agros, aedificiorum sola, ex quibus altitudo aut crassitudo non proponitur, ut opera tectoria, inauraturas, tabulas, et his similia. Solidum est quod Graeci stereon appellant, nos quadratos pedes appellamus; cuius longitudinem et latitudinem et crassitudinem metimur, ut parietum structuras, pilarum pyramidum aut lapidum materias, et his similia.

A plane is that which the Greeks call epipedon, which we call constratos pedes; in which we have length and breadth; by which we measure fields, the flat surfaces of buildings alone, from which height or thickness is not set forth, as roof-works, gildings, boards, and the like. A solid is that which the Greeks call stereon, which we call quadratos pedes; whose length and breadth and thickness we measure, as the structures of walls, the materials of pillars, pyramids or stones, and the like.

[9] Omnis autem mensurarum observatio et oritur et desinit signo. Signum est cuius pars nulla est. Haec est omnium extremitatium finitima contemplatio.

[9] Every observation of measurements both begins and ends with a sign. A sign is that of which there is no part. This is the most final contemplation of all extremities.

Signum autem sine parte est initium, a quo omnia incipiunt. Extremitas est quo usque uni cuique possidendi ius concessum est, aut quo usque quisque suum servat. Extremitatium genera sunt duo, unum quod per rigorem observatur, alterum quod per flexus.

A sign, however, without part, is the beginning from which all things commence. An extremity is how far to each one the right of possessing has been granted, or how far each preserves his own. The kinds of extremities are two: one which is observed by rigor, the other which by flexure.

Rigor est quidquid inter duo signa veluti in modum lineae rectum perspicitur; per flexus, quidquid secundum locorum naturam curvatur, ut in agris archifiniis solet. Decumanus est longitudo rationalis, itemque cardo, constitutis in unum binis rigoribus, singulis spatio itineris interveniente.

Rigor is whatever between two signs is perceived as straight, as it were in the manner of a line; by flexes, whatever is curved according to the nature of places, as is customary in the arch-boundaries of fields. Decumanus is the rational length, and likewise the cardo, the two rigors being constituted together into one, each with an intervening space of journey.

[10] Nam quidquid in agro mensorii operis causa ad finem rectum fuerit, rigor appellatur: quidquid ad horum imitationem in forma scribitur, linea appellatur. Linea est longitudo sine latitudine, lineae autem fines signa. Ordinatae rectae lineae sunt quae in eadem planitia positae et eiectae in utramque partem in infinitum non concurrunt.

[10] For whatever in a field for the purpose of surveying work is straight to an end is called rigor; whatever is written in form in imitation of these is called a line. A line is length without breadth, and the ends of lines are signs. Ordered straight lines are those which, placed in the same plane and thrown out to either side to infinity, do not meet.

[11] Linearum genera sunt trea, rectum, circum ferens, flexuosum. Recta linea est quae aequaliter suis signis rectis posita est; circum ferens, cuius incessus a conspectu signorum suorum distabit. Flexuosa linea est multiformis, velut arvorum aut iugorum aut fluminum; in quorum similitudinem et arcifiniorum agrorum extremitas finitur, et multarum rerum similiter, quae natura inaequali linea formata sunt.

[11] The kinds of lines are three: straight, circumferent, and flexuous. A straight line is that which is placed equally by its right signs; a circumferent line is one whose course will be distant from the sight of its signs. A flexuous line is manifold, like of fields or of ridges or of rivers; in the likeness of which the extremity of archifinium fields is bounded, and likewise of many things which by nature are formed with an unequal line.

[12] Summitas est secundum geometricam appellationem quae longitudinem et latitudinem tantum modo habet, summitatis fines lineae. Plana summitas est quae aequaliter rectis lineis est posita. Omnium autem summitatium metiundi observationes sunt duae, enormis et liquis; enormis, quae in omnem actum rectis angulis continetur; liquis, quae minuendi laboris causa et salva rectorum angulorum ratione secundum ipsam extremitatem subtenditur.

[12] A summitas is, according to geometric appellation, that which has only length and latitude; the limits of a summitas are lines. A plane summitas is that which is laid equally by straight lines. The rules for measuring all summitats are two, the enormis and the liquis: the enormis, which is contained in every extent by right angles; the liquis, which is subtended to the very extremity for the sake of lessening the labor of measurement and preserving the rationale of the right angles.

[13] Genera angulorum rationalium sunt tria, rectum ebes acutum. Haec habent species VIIII; rectarum linearum tres, rectarum et circumferentium tres, circumferentium tres. Rectarum ergo linearum species angulorum generis sui tres, recta ebes acuta.

[13] The kinds of rational angles are three: right, obtuse, acute. These have nine species; three of straight lines, three of straight and circumferential, three of circumferential. Therefore the species of angles of the genus of straight lines are three: right, obtuse, acute.

Rectus angulus est euthygrammos, id est ex rectis lineis conprehensus, qui Latine normalis appellatur. Quotiens autem recta super recta linea stans ex ordine angulos pares fecerit, et singuli anguli recti sunt, et stans perpendicularis eius lineae super quam insistit est. Cuius sede si subtendens linea perpendiculari fuerit iniuncta, efficit triangulum recto angulo.

A right angle is euthygrammos, that is, composed of straight lines, which in Latin is called normal. Whenever a straight line standing upon another straight line in order makes equal angles, each of the angles is right, and the standing line is perpendicular to the line upon which it rests. From whose seat, if a subtending line be drawn perpendicular to it, it produces a triangle with a right angle.

ebes angulus est plus normalis, hoc est excedens recti anguli positionem, et qui, si triangulus secundum hanc positionem constitutus fuerit, perpendicularem extra finitimas lineas habeat. Acutus angulus est conpressior recto; qui si a recta linea, quae sedis loco fuerit, rectam lineam secundum suam inclinationem emiserit, similique cohibitione rectam lineam in occursum exceperit, efficiet triangulum qui perpendicularem intra tres lineas habebit. Rectus ergo angulus est normalis, ebes plus normalis, acutus minus normalis.

ebes angle is the more normal, that is, exceeding the position of the right angle, and which, if a triangle be constructed according to this position, will have the perpendicular outside the bounding lines. The acute angle is more compressed than the right; which, if from the right line that serves as the seat a straight line be sent out according to its inclination, and by a like constraint the straight line meet it coming in encounter, will produce a triangle that has the perpendicular within the three lines. Therefore the right angle is normal, the ebes more normal, the acute less normal.

Rectarum linearum et circumferentium species angulorum generis sui tres, recta ebes acuta. Quaecumque autem linea in dimensione medium secans circulum per punctum transiens ad circumferentem lineam pares alternos secundum suam speciem rectos angulos faciet. Ebetes angulos faciet generis sui quaecumque ordinata dimensioni linea intra semicirculum, in eo tamen spatio quod inter se et lineam quae per punctum semicirculi transiet interiacebit.

There are three species of angles of right lines and circumferences of their kind: right, obtuse (ebes), and acute. Whatever line, in a median dimension cutting the circle and passing through a point to the circumferential line, will make alternate equal right angles according to its kind. An obtuse (ebes) will be made by any line ordinate to the diameter within the semicircle, yet in that space which will lie between itself and the line that passes through the point of the semicircle.

Quotiens intra semicirculum linea fuerit ordinata dimensionis lineae, acutos angulos faciet generis sui, quos in circumferentia cludet. Rectarum ergo et circumferentium linearum anguli rectus ebes acutus; rectus, quoniam recta linea quae per punctum ad circumferentiam pervenit, medium secat circulum et utraque parte pares angulos dividit; ebes et acutus ideo quod ordinata dimensioni linea intra semicirculum inferiores facit angulos maiores: nam quos intra circumferentiam cludet, minores.

Whenever an ordinate of the dimension line is placed within the semicircle, it will make acute angles of its kind, which it will enclose on the circumference. Hence the angles of straight and circumferential lines are right, ebes, acute; right, because a straight line which, passing through the point, reaches the circumference, cuts the circle through the middle and divides equal angles on each side; ebes and acute, therefore, because the ordinate to the dimension line within the semicircle, being lower, makes the angles larger: for those which it will enclose within the circumference are smaller.

[14] Circumferentium linearum species angulorum generis sui tres, recta ebes acuta. Quotiens ex uno duorum punctorum diastemate duo circuli pares exeunt, ad conexionem circumferentiarum interiores rectos angulos facient; ebetes exteriores, qui sunt sescontrarii rectis: acuti anguli sunt lunati, qui inter rectos et ebetes includuntur. Circumferentium linearum rectos angulos ideo quod si tres circuli pares inter se fuerint aequali diastemate conexi, intra scriptos angulos pares alternos habebunt, per quorum signa si rectae lineae intra scribantur, in partes quas circulorum conexio consumet medias divident.

[14] The kinds of angles of circumferential lines are three of their genus: right, obtuse, and acute. Whenever, from the separation of one of two points, two equal circles proceed, at the connection of the circumferences the interior ones will make right angles; the exterior ones obtuse, which are sescontrary to the right; the acute angles are lunate, which are enclosed between the right and the obtuse. The circumferential lines are called right-angled therefore because if three equal circles are joined to one another at equal spacing, they will have equal alternate angles within the described angles; and by those marks, if straight lines are drawn within, they will divide the parts which the connection of the circles occupies into middle portions.

Ebetes angulos exteriores, quod sunt omnibus intra scriptis maiores. Lunati autem acuti, quod exilissima tenuitate finiuntur.

The obtuse are the exterior angles, because they are greater than all those enclosed within the written (circumference). The lunate, however, are the acute, because they are terminated by the most slender thinness.

[15] Rationalium linearum genera angulorum haec sunt. Quibus si flexuosa linea iniungatur, faciet species angulorum secundum suam inaequalitatem complures: omnes tamen illae inaequalitates rationalibus lineis conprehendi et dividi possunt. [Flexuosa autem linea sicut elicis aut cornualis.] Nam flexuosa linea ad mensuram redigitur, quem admodum ipsius loci natura permittit, qua proxima est rectae lineae adque circumferenti circulari, si terminibus arboribus notatis aut fossis aut viis aut iugis montium et divergiis aquarum fines observabuntur.

[15] These are the kinds of angles of rational lines. If to these a flexuous line is attached, it will produce several species of angles according to its inequality: yet all those inequalities can be comprehended and divided by rational lines. [A flexuous line, moreover, is like a helix or horn-shaped (elicis aut cornualis).] For a flexuous line is reduced to measure insofar as the nature of the place itself permits, by how near it is to the straight line and to the circumferential circle, if by marked termini — trees or ditches or ways or mountain ridges and divergences of waters — the bounds are observed.

[16] Angulus autem omnis species capit duas, planam et solidam. Planus angulus est in planitia duarum linearum adtingentium, sed et non in rectum positarum, alterius ad alteram inclinatio. Solidus angulus est cuius planitiae altitudo adiungitur aut aequatur.

[16] Now every kind of angle comprises two: the plane and the solid. A plane angle is the inclination of one to another of two lines meeting in a plane, but not placed at right angles to one another. A solid angle is one to which the altitude of the plane is added or made equal.

[17] Forma est quae sub aliquo aut aliquibus finibus continetur. Formarum genera sunt quinque. Vnum quod ex flexuosa linea continetur.

[17] A form is that which is contained under some or certain bounds. The genera of forms are five. One (unum) is that which is contained by a flexuous line.

Alterum quod ex flexuosa et rationalibus. tertium quod ex circumferentibus. Quartum quod ex circumferentibus et rectis.

The second which is of flexuous and rational lines. the third which is of circumferent lines. Fourth which is of circumferent and straight lines.

Quintum quod ex rectis. Horum generum sunt species multitudinis infinitae. Flexuosarum linearum formae species habent multas in infinitum.

The fifth is that which is from straight lines. The species of these genera are of infinite multitude. The species of forms of flexuous lines have many unto infinity.

Aeque multas ac varias figuras habent formae, quotiens flexuosae lineae rationalis sive recta sive circularis linea intervenit.

Aeque forms have many and various figures, whenever a rational line, whether straight or circular, intervenes among winding lines.

[18] Circumferentium linearum formae aliquae sunt sine angulo, aliquae uno, aliquae duorum, aliquae trium, aliquae quattuor, et aliquae super hunc numerum singulis angulis accedentibus ut plurimum in infinitum. Forma est sine angulo circuli unius pluriumve. Circulus autem est plana forma ab una linea conprehensa, ad quam ab uno signo intra formam posito omnes accedentes rectae lineae sunt inter se pares.

[18] The forms of circumferential lines are some without an angle, some with one, some with two, some with three, some with four, and some above this number, single angles being added, for the most part to infinity. A form without an angle is a circle of one or of many. A circle, however, is a plane form bounded by one line, to which, a single point placed within the form, all straight lines drawn to it are equal among themselves.

Ex pluribus circulis forma sine angulo, ut harenae ex quattuor circulis; ex pluribus quam quinque, ut in opere picturarum aut architectura. forma anguli unius ex tribus circinis, ut in opere marmoreo. Duorum angulorum forma e duobus circinis, trium angulorum ex tribus circinis, quattuor angulorum ex quattuor circinis, reliquae accedentibus singulis plurilaterae in infinitum.

From several circles a form without angle, as the sand from four circles; from several more than five, as in the work of paintings or architecture. The form of one angle from three circles, as in marble work. The form of two angles from two circles, of three angles from three circles, of four angles from four circles, the rest, with each single one added, many-sided to infinity.

Rectarum linearum et circumferentium [forma sine angulo] duorum laterum totidemque angulorum forma est ex recta linea et circumferenti semicirculo. [Rectarum linearum et circumferentium formae sine angulo lateris unius, duorum angulorum ex duobus lateribus, trium angulorum ex tribus lateribus, quattuor ex quattuor, reliquae singulis accedentibus plurilaterae.] Trilatera forma est trium laterum totidemque angulorum ex duabus rectis lineis et una circumferenti, vel ex duabus circumferentibus et una recta. Ex duabus ergo rectis et una circumferenti.

The form without angle of straight lines and circumferentials of two sides and of as many angles is from a straight line and a semicircle as circumferential. [The forms without angle of straight lines and circumferentials of one side, of two angles from two sides, of three angles from three sides, of four from four, the remaining multilateral with single ones adjoined to infinity.] A trilateral form is of three sides and as many angles from two straight lines and one circumferential, or from two circumferentials and one straight. Therefore from two straight lines and one circumferential.

Ex duabus circumferentibus et recta. Quadrilatera forma est quattuor laterum totidemque angulorum ex quattuor lineis comprehensa, ut duabus rectis et duabus circumferentibus. Plurilatera forma est quae plus quam quattuor lineis comprehensa est, ut quinque laterum totidemque angulorum ex duabus rectis et tribus circumferentibus, ex tribus rectis et duabus circumferentibus.

From two circumferentials and a straight line. A quadrilateral form is four sides and as many angles encompassed by four lines, as, for example, by two straight lines and two circumferentials. A plurilateral form is that which is encompassed by more than four lines, as five sides and as many angles formed from two straight lines and three circumferentials, or from three straight lines and two circumferentials.

Et quaecumque huic formae accedentibus singulis angulis et lateribus similis fuerit, plurilatera appellatur.

And whatever is similar to this form when each of the angles and sides is added to it is called plurilateral.

[21] Planarum autem et rectis lineis comprehensarum aliae sunt trilaterae, aliae quadrilaterae, aliae singulis adiectis super hunc numerum plurilaterae in infinitum. Trilatera forma est quae tribus rectis lineis continetur. Trilaterarum formarum et ex rectis lineis comprehensarum species sunt quattuor.

[21] Of plane figures enclosed by straight lines, some are trilateral, others quadrilateral, others, with single lines added beyond this number, multilateral to infinity. A trilateral form is that which is contained by three straight lines. The species of trilateral forms composed of straight lines are four.

Vna qua rectus angulus continetur, et efficit triangulum recto angulo, quod Graeci orthogonion appellant. . . .

One in which a right angle is contained, and which produces a triangle with a right angle, which the Greeks call orthogonion . . .

[20] Plurilatera forma est quae plus quam quattuor rectis lineis sub qualicumque specie continetur . . .

[20] A plurilateral form is that which is contained by more than four straight lines under any kind of figure . . .

. . .

Quinque, quam formam Graeci pentagonon appellant. Amplioribus quoque formis apud Graecos nomina ab angulis dantur, ut hexagono heptagono et super hunc numerum compluribus. Has nos plurilateras appellamus adiecto angulorum numero, ut sex angulorum et septem.

Five, which form the Greeks call pentagon. To the larger forms likewise among the Greeks names are given from the angles, as hexagon, heptagon, and for numbers above this many more. We call these multilaterals, adding the number of angles, as six-angled and seven-angled.

Et quantumcumque super hunc numerum auxeris, eandem appellationem utamur.

And however much you increase beyond this number, let us use the same appellation.

[21] Alia species est formae per quam frequenter archifiniorum agrorum quadratura concluditur ex rectis angulis [ex] pluribus quam quinque, accedentibus super hunc numerum in quantacumque multitudine cogitaveris. [Qualemcumque rectorum angulorum formam rectis lineis comprehendere.

[21] Another kind is the form by which the quadrature of arable fields bounded by straight lines is frequently concluded from right angles, from more than five, the angles exceeding this number in whatever multitude you may conceive. To comprehend by straight lines whatever shape of right angles.

[22] Ex data recta linea ducere posito signo . . .

[22] From a given straight line to draw, a mark having been placed . . .

. . . Relato in utramque partem circino, aequali punctorum diastemate circulos scribere oportet, per quorum conexionem recta linea transeat factura normales in data linea angulos. Sed quo in rectarum linearum forma circularis linea non interveniat rectis, a circumferentiarum parte chiasmi cuiusdam ratione utamur.

. . . With the compass carried to either side, one ought to draw circles with an equal diastema of points, through whose connexion a straight line will pass, making normals at the angles in the given line. But where a circular line does not intervene among the forms of straight lines, we employ, from the part of the circumferences, a certain chiasm method.

[23] Quod si ab eadem recta linea ducenda fuerit quae rectum angulum faciat, ex quolibet puncto qui per caput recta linea transeat rectam lineam eicere, per cuius signum quod est in circumferentem lineam a capite rectae lineae recta linea transeat factura in data linea rectum angulum.

[23] But if from the same straight line to be drawn there is one that makes a right angle, then from any point through which a straight line passing through the head crosses, draw a straight line; by whose mark, which lies on the circumferential line, a straight line drawn from the head of the straight line will cross the given line at a right angle.

[24] In hanc autem rationem sublata circumferentia chiasmis utendum est. Nam quod ad extremam lineae normationem pertinet, vulgaris consuetudinis est sex octo et decem: haec de qua supra disputavimus circuli ratio magis artificialis est, quae numeros non praefinit: habemus enim apud Eucliden, quocumque loco ad circumferentem lineam ex signis dimensionis duae lineae concurrerint, normam facturas.]

[24] Into this method, with the circumference removed, a chiasm is to be used. For as to the normalization of the line's extremity, the common custom is six, eight, and ten: this circle-ratio of which we argued above is more artificial, and does not preset numbers; for we have in Euclid that wherever, at any place on the circumference line, two lines from the signs of measurement have met, they will make the norm.