§1.10.37iam primum ordo est geometriae necessarius; nonne et eloquentiae? ex prioribus geometria probat insequentia, ex certis incerta; nonne id in dicendo facimus? quid? illa propositarum quaestionum conclusio non fere tota constat syllogismis? propter quod plures invenias, qui dialecticae similem quam qui rhetoricae fateantur hanc artem.
Now, first of all, order is necessary to geometry; is it not also to eloquence? From prior things geometry proves what follows, from certain things what is uncertain; do we not do this in speaking? What? Is not that conclusion of the proposed questions almost entirely made up of syllogisms? For which reason you would find more who confess this art is similar to dialectic than to rhetoric.
verum et orator etiamsi raro non tamen nunquam probabit dialectice.
But even the orator, although rarely, yet not never, will prove dialectically.
§1.10.38nam et syllogismo, si res poscet, utetur et certe enthymemate, qui rhetoricus est syllogismus.
For he will also use a syllogism, if the matter demands, and certainly an enthymeme, which is a rhetorical syllogism.
denique probationum quae sunt potentissimae γραμμικαὶ ἀποδείξεις vulgo dicuntur: quid autem magis oratio quam probationem petit?
Finally, those proofs which are the most powerful are commonly called "linear proofs": but what does speech seek more than proof?
§1.10.39falsa quoque veris similia geometrica ratione deprehendit.
Geometry also detects falsehoods similar to truths by geometric reasoning.
fit hoc et in numeris per quasdam, quas ψευδογραφίας vocant, quibus pueri ludere solebamus.
This happens also in numbers through certain things which they call "pseudographiae", with which we as boys used to play.
sed alia maiora sunt.
But there are other greater things.
nam quis non ita proponenti credat?
"quorum locorum extremae lineae eandem mensuram colligunt, eorum spatium quoque, quod iis lineis continetur, par sit necesse est. "
For who would not believe one proposing thus: "Of which places the outermost lines gather the same measure, of those the space also which is contained by those lines must of necessity be equal"?
§1.10.40at id falsum est.
But that is false.
nam plurimum refert, cuius sit formae ille circuitus; reprehensique a geometris sunt historici, qui magnitudinem insularum satis significari navigationis ambitu crediderunt.
For it matters very much of what shape that circuit is; and historians have been censured by geometers, who believed that the magnitude of islands was sufficiently indicated by the circuit of navigation.
nam ut quaeque forma perfectissima ita capacissima est.
For as each shape is most perfect, so it is most capacious.
§1.10.41ideoque illa circumcurrens linea si efficiet orbem, quae forma est in planis maxime perfecta, amplius spatium complectetur quam si quadratum paribus oris efficiat, rursus quadrata triangulis, triangula ipsa plus aequis lateribus quam inaequalibus.
And therefore if that running-around line shall make a circle, which is the most perfect shape in planes, it will embrace a larger space than if it should make a square with equal boundaries; again, squares embrace more than triangles, and triangles themselves more with equal sides than with unequal.
§1.10.42sed alia forsitan obscuriora; nos facillimum etiam imperitis sequamur experimentum.
But other things are perhaps more obscure; let us follow an experiment very easy even to the unskilful.
lugeri mensuram ducentos et quadraginta longitudinis pedes esse dimidioque in latitudinem patere, non fere quisquam est qui ignoret, et qui sit circuitus et quantum campi claudat, colligere expeditum.
That the measure of a iugerum is two hundred and forty feet in length, and lies open in width by half, there is almost no one who is ignorant, and it is easy to calculate both what the circuit is and how much of a field it encloses.
§1.10.43at centeni et octogeni in quamque partem pedes idem spatium extremitatis sed multo amplius clausae quattuor lineis areae faciunt.
But one hundred and eighty feet on each side make the same space of boundary, but a much larger area enclosed by the four lines.
id si computare quem piget, brevioribus numeris idem discat.
If anyone is loath to calculate this, let him learn the same by smaller numbers.
nam deni in quadram pedes, quadraginta per oram, intra centum erunt.
For ten feet each way into a square, forty along the boundary, will be within a hundred.
at si quini deni per latera, quini in fronte sint, ex illo, quod amplectuntur, quartam deducent eodem circumductu.
But if there are fifteen on the sides and five in front, they will subtract a fourth part from that which they embrace, with the same circuit.
§1.10.44si vero porrecti utrinque undeviceni singulis distent, non plures intus quadratos habebunt, quam per quot longitudo ducetur; quae circumibit autem linea, eiusdem spatii erit, cuius ea quae centum continet.
If indeed stretched out on each side to nineteen they are distant by single feet, they will not have more squares inside than that number by which the length is multiplied; but the line which goes around will be of the same length as that which contains a hundred.
ita quidquid formae quadrati detraxeris, amplitudini quoque peribit.
Thus, whatever you subtract from the shape of the square will also be lost to its amplitude.
§1.10.45ergo etiam id fieri potest, ut maiore circuitu minor loci amplitudo claudatur.
Therefore it can also happen that a smaller amplitude of space is enclosed by a larger circuit.
haec in planis.
These things in planes.
nam in collibus vallibusque etiam imperito patet plus soli esse quam caeli.
For in hills and valleys it is clear even to the unskilful that there is more of ground than of sky.
§1.10.46quid quod se eadem geometria tollit ad rationem usque mundi? in qua, cum siderum certos constitutosque cursus numeris docet, discimus nihil esse inordinatum atque fortuitum; quod ipsum nonnunquam pertinere ad oratorem potest.
What of the fact that this same geometry raises itself up even to the system of the universe? In which, when it teaches the certain and established courses of the stars by numbers, we learn that nothing is disordered and fortuitous; which very thing can sometimes pertain to the orator.
§1.10.47an vero, cum Pericles Athenienses solis obscuratione territos redditis eius rei causis metu liberavit, aut cum Sulpicius ille Gallus in exercitu L. Paulli de lunae defectione disseruit, ne velut prodigio divinitus facto militum animi terrerentur, non videtur usus esse oratoris officio?
Or indeed, when Pericles freed the Athenians, terrified by an eclipse of the sun, from fear by explaining the causes of that thing, or when that famous Sulpicius Gallus in the army of Lucius Paullus discoursed on the eclipse of the moon, lest the minds of the soldiers should be terrified as if by a prodigy divinely caused, does he not seem to have performed the office of an orator?
§1.10.48quod si Nicias in Sicilia scisset, non eodem confusus metu pulcherrimum Atheniensium exercitum perdidisset; sicut Dion, cum ad destruendam Dionysii tyrannidem venit, non est tali casu deterritus.
Which if Nicias had known in Sicily, he would not, confused by the same fear, have destroyed the most beautiful army of the Athenians; just as Dion, when he came to destroy the tyranny of Dionysius, was not deterred by such an event.
sint extra licet usus bellici, transeamusque, quod Archimedes unus obsidionem Syracusarum in longius traxit.
Let military utilities be permitted to be outside, and let us pass over the fact that Archimedes alone prolonged the siege of Syracuse.
§1.10.49illud utique iam proprium ad efficiendum quod intendimus, plurimas quaestiones, quibus difficilior alia ratione explicatio est, ut de ratione dividendi, de sectione in infinitum, de celeritate augenda, linearibus illis probationibus solvi solere; ut, si est oratori (quod proximus demonstrabit liber) de omnibus rebus dicendum, nullo modo sine geometria esse possit orator.
That, at all events, is now proper to effect what we intend, that very many questions, of which the explanation by other means is more difficult, as concerning the method of dividing, concerning section to infinity, concerning the increasing of speed, are accustomed to be solved by those linear proofs; so that, if the orator must speak of all things (as the next book will demonstrate), the orator can in no way exist without geometry.