ELEMENTS

OF

GEOMETRY.

BOOK VI.

THE doctrine of Proportion, grounded on the simplest theory of numbers, furnishes a most powerful instrument, for abridging and extending mathematical investigations. It easily unfolds the primary relations of figures, and the sections of lines and circles; but it also discloses with admirable felicity that vast concatenation of general properties, not less important than remote, which, without such aid, might for ever have escaped the penetration of the geometer. He is thus placed on a commanding eminence; from which he views the bearings of the objects below, surveys the contours of the distant amphitheatre, and descries the fading

verge of a boundless horizon. The application of Arithmetic to Geometry forms, therefore, one of those grand epochs which occur, in the lapse of ages, to mark and accelerate the progress of scientific discovery.

DEFINITIONS.

1. Straight lines which proceed from the same point, are termed diverging lines.

2. Straight lines are divided similarly, when their corresponding segments have all the same ratio.

3. A straight line is said to be cut harmonically, if it consist of three segments, such that the whole line is to one extreme, as the other extreme to the middle part.

4. The area of a figure is its surface, or the quantity of space which it occupies.

5. Similar figures are snch as have their angles respectively equal, and the containing sides proportional.

6. If two sides of a rectilineal figure be the extremes of an analogy, of which the means are two sides containing an equal angle in another rectilineal figure; these sides are said to be reciprocally proportional.

PROP. I. THEOR.

Parallels cut diverging lines proportionally.

The parallels DE and BC cut the diverging lines AB and AC into proportional segments.

Those parallels may lie on the same side of the vertex, or on opposite sides; and they may consist of two, or of more lines.

1. Let the two parallels DE and BC intersect the diverging lines AB and AC, on the same side of the vertex A; then are AB and AC cut proportionally, in the points D and E,-or AD: AB:: AE: AC.

For if AD be commensurable with AB, find (V. 25.) their common measure M, and, from the corresponding points of section in AD and AB, draw (I. 26.) the parallels FI, GK, and HL. It is evident, from Book I. Prop. 40, that these parallels will also divide the straight lines AE and AC equally. Wherefore the measure M, or AF the submultiple of AD, is contained in AB, as often as AI, the like submultiple of AE, is contained in AC; consequently (V. def. 10) the ratio of AD to AB is the same with that of AE to AC.

T

K

E

A F GDH B

But, should the segments AD and AB be incommensurable, they may still be expressed numerically, and that to any required degree of precision. AD being divided into equal parts (I. 40.), these parts, continued towards B, will, together with a subsidiary portion, compose the whole of

M

AB. Let this division of AD extend in DB to b, and draw the parallel bc. If the parts of AD and AB be again subdivided, the corresponding residue will evidently be diminished; and thus, at each successive subdivision, the terminating parallel bc must approximate perpetually to BC. Wherefore, by continuing this process of exhaustion, the divided lines Ab and Ac will approach the limits AB and AC, nearer than any finite or assignable interval. Consequent

E

D bB

ly, from the preceding demonstration, AD: AB :: AE: AC.

And since AD: AB:: AE: AC, it follows, by conversion (V. 11.), that AD: DB:: AE: EC, and again, by composition (V. 9.), that AB : DB:: AC : EC.

2. Let the two parallels DE and BC cut the diverging lines DB and EC, on opposite sides of A; the segments AB, AD have the same ratio with AC, AE,-or AB: AD: AC: AE.

.

P

For, make AO equal to AD, AP to AE, and join OP. The triangles APO and AED, having the sides AO, AP equal to AD, AE, and the contained vertical angle OAP equal to DAE, are equal (I. 3.), and consequently the angle AOP is equal to ADE; but these being alternate angles, the straight line OP (I. 25.) is

D

A

B

parallel to DE, and hence, from what was already demonstrated, AB: AO or AD:: AC: AP or AE.

And since AB: AD: AC: AE, by conversion DB: DA:: EC: EA, and, by conversion, and inversion DB: AB:: EC: AC.