THEOREM XXI.

If the product of two quantities be equal to that of two others, they are reciprocally proportional.

Let P be the product of A and D; let Q be that of B and C, and let P be equal to Q.

Let D be equal to m units; then P is equal to m times A [Gen. Def. 37]. Because Q is the product of C by B [xix], the product Q is to C as B is to one unit [xviii]; therefore P is also to C as B is to one unit: now B is to the unit as m times B is to m units, or D [xi]; then P is to C as m times B is to D [i]; therefore P is to m times B as C is to D [xiii]. Because A is to B as m times A is to m times B [xi], and m times A are equal to P, the quantity A is to B as P is to m times B; therefore A is to B as C is to D [i], which W. T. B. D.

THEOREM XXII.

If two continual proportions have the extremes of the one equal to those of the other, each to each, the means of the one are equal to those of the other.

Let A be to B as B is to D, and let A be to C as C is to D.

Because the product of B by B and that of C by C are each equal to that of A by D [xx], the product of B by B is equal to that of C by C [i]; thence B is to C as C is to B [xxi]: but B cannot be to C as O is to B unless the ratio of the corresponding terms be equal to one; therefore B is equal to C, which W. T. B. D.

PLANE GEOMETRY

IN FOUR BOOKS.

BOOK I.

ON LINES.

Since the classifications in any Science are continually modified as scientific knowledge advances, the definitions in the sciences are also varying."

MILL.

PLANE GEOMETRY.

BOOK I.

ON LINES.

PRELIMINARY DEFINITIONS:

I. LINES.

"A straight line is that of which the extremity hides all the rest, the eye being placed in the continuation of the line." PLATO.

"A straight line is that which lies similarly between its extreme points." EUCLID.

"A straight line is the shortest distance between two points." ARCHIMEDES.

Let there be imagined three points, the third of which is placed in the same position relatively to both the first and the second as the second is placed relatively to the first, the distance being disregarded, so that the figure formed by the three points be similar [Gen. Def. 19] to that formed by the first two; then let there be imagined a fourth point placed relatively to each of the preceding ones in the same position as the third is relatively to each of those preceding it, and so on indefinitely; the idea of a line will thus be conceived, all points of which will proceed in the same way, and which will differ in magnitude only from the figure formed by the two first points, or any two of the points: this line is called a straight line. Thence

I. A straight line is that which is similar to each of its parts.1

It is evident that a straight line may have any portion thereof expanded, or reduced, without ceasing to be straight, for then it remains similar to

1 In order the better to understand this definition, and all others of the same kind, the line must be imagined as great as it can be; that is, infinite.

D.