BOOK ONE

OF THE PERSPECTIVE OF WITELO.

DEFINITIONS.

Those things which we set forth as principles for this first book are these. We call a cathetus a perpendicular line erected upon some surface. We call a pole every point of a line erected orthogonally from the center above the surface of a circle. We call a line or surface convex which has some regular curvature on the outside. We call a line or surface concave which has some regular curvature on the inside. We call a line perpendicular to a convex or concave surface which is erected upon a plane surface touching the convex or concave surface at the point of its incidence. Circles are said to be intersecting one another when there is some line common to their diameters, one not containing the other. A great circle of a sphere is said to be that which, passing through the center of the sphere, divides it into two equal parts. A lesser circle of a sphere is said to be that which neither passes through the center of the sphere nor divides it into two equal parts. We call spheres equal whose diameters are equal. We call spheres or circles containing one another equidistant [concentric], between which lines drawn from the center of the larger, from the convex of the smaller to the concave of the larger, are equal. We call spheres tangent to one another which, touching each other externally or internally, do not intersect. We call spheres intersecting one another when, the spheres not containing each other, the diameter of one is cut through by the other. We call spheres intersecting internally those in which the greater part of one is contained in the other. We say a plane surface touches a sphere when, though it touches the sphere, it does not cut it when extended in any direction. The denomination of a ratio of the first to the second is said to be the quantity which, multiplied by the smaller, produces the larger, or which divides the larger by the smaller. A ratio is said to be composed of two ratios when the denomination of that ratio is produced from the multiplication of the denominations of those ratios, one by the other.

PETITIONS.

A geometric diagram on the left margin showing three lines originating from a common vertex at the bottom, labeled 'b'. The lines extend upwards and outwards. The leftmost line is labeled 'a' near the top, with a point 'e' marked further up. The middle line is labeled 'c' at the top. The rightmost line is labeled 'd' at the top.

e
c
a
b

Moreover, we petition [postulate] these things: That equal angles constituted upon the same point contain an equal distance of equal lines; so that if angles $abc$ and $cbd$ are equal, and lines $ab$ and $bd$ are equal, line $ab$ will be as distant from line $bc$ as line $bd$ is distant from the same line $bc$. Likewise, that a line can be extended between any two points, and a surface between any two lines. Likewise, that when two plane surfaces touch each other, they become one surface. Likewise, that two plane surfaces do not enclose a body. Likewise, that all the same ratios are composed of similar ratios, and divided into similar ratios, and have the same demonstrations.

THEOREMA I.

All parallel lines necessarily consist in the same plane surface.

A geometric diagram on the left margin consisting of two vertical parallel lines. The left line has endpoints labeled 'A' at the top and 'B' at the bottom. The right line has endpoints labeled 'C' at the top and 'D' at the bottom. A horizontal line segment connects point 'B' on the left to point 'D' on the right.

A
B
C
D

Let there be two parallel lines $ab$ and $cd$ disposed in any way; I say that they are in the same plane surface. For let them be joined by the line $bd$; since therefore lines $ab$ and $bd$ are joined angularly, it is clear that they are in the same surface, by the second [proposition] of the eleventh [book of Euclid]. Similarly, because the two lines $ad$ and $bd$ are joined angularly, they will be in the same surface. If line $bd$ is in only one plane surface, then since it is impossible for part of it to be in the air and part in the plane, by the first [proposition] of the eleventh [book of Euclid], it is clear, therefore, that lines $ab$ and $cd$ necessarily consist in the same plane surface.