Now in regard to these two portions of the conceivable or intelligible region, two different procedures of the mind are employed: the pure Dialectic, and the Geometrical, procedure. The Geometer or the Arithmetician begins with certain visible images, lines, figures, or numbered objects, of sense: he takes his departure from certain hypotheses or assumptions, such as given numbers, odd and even--given figures and angles, of three different sorts.[201] He assumes these as data without rendering account of them, or allowing them to be called in question, as if they were self-evident to every one. From these premisses he deduces his conclusions, carrying them down by uncontradicted steps to the solution of the problem which he is examining.[202] But though he has before his eyes the visible parallelogram inscribed on the sand, with its visible diagonal, and though all his propositions are affirmed respecting these--yet what he has really in his mind is something quite different--the Parallelogram _per se_, or the Form of a Parallelogram--the Form of a Diagonal, &c. The visible figure before him is used only as an image or representative of this self-existent form; which last he can contemplate only in conception, though all his propositions are intended to apply to it.[203] He is unable to take his departure directly from this Form, as from a first principle: he is forced to assume the visible figure as his point of departure, and cannot ascend above it: he treats it as something privileged and self-evident.[204] [Footnote 201: Plato, Republic, vi. p. 510 B. [Greek: ê(=| to me\n au)tou= (tmê=ma) toi=s to/te tmêthei=sin ô(s ei)ko/si chrôme/nê] (this is farther illustrated by p. 511 A--[Greek: ei)ko/si chrôme/nên au)toi=s toi=s u(po\ tô=n ka/tô a)peikasthei=si) psuchê\ zêtei=n a)nagka/zetai e)x u(pothe/seôn, ou)k e)p' a)rchê\n poreuome/nê a)ll' e)pi\ teleutê/n], &c.] [Footnote 202: Plato, Republic, vi. p. 510 C-D. [Greek: oi( peri\ ta\s geômetri/as te kai\ logismou\s kai\ ta\ toiau=ta pragmateuo/menoi, u(pothe/menoi to/ te peritto\n kai\ to\ a)/rtion kai\ ta\ schê/mata kai\ gôniô=n tritta\ ei)/dê kai\ a)/lla tou/tôn a)delpha\ kath' e(ka/stên me/thodon, tau=ta me\n ô(s ei)do/tes, poiêsa/menoi u(pothe/seis au)ta/, ou)de/na lo/gon ou)/te au(toi=s ou)/te toi=s a)/llois e)/ti a)xiou=si peri\ au)tô=n dido/nai, ô(s panti\ phanerô=n; e)k tou/tôn d' a)rcho/menoi ta\ loipa\ ê)/dê diexio/ntes teleutô=sin o(mologoume/nôs e)pi\ tou=to, ou)= a)\n e)pi\ ske/psin o(rmê/sôsin.]]