If any equimultiples of the terms of one of the ratios of an analogy be taken, and also any equimultiples of the terms of the other ratio, these equimultiples taken in the order of the terms are proportional.

Let A B C :D, then m.A: mBnC: nD.

=

For A: B=mA: mB (20), and C: DnC:nD; and hence mA: mBnC: nD.

If any equimultiples be taken of two homologous terms of an analogy, and also any equimultiples of the other two terms, these multiples taken in the order of the terms are proportional.

Let A:B= C: D, then mA : nB = mC : nD

For, by alternation, A: CB: D, and therefore (22) mA: mCnB: nD, or, by alternation, mA : nB = mC: nD.

Schol. The last three propositions are true whether m or n be terminate or interminate numbers.

If two terms that are not homologous in one analogy be equal to two that are not homologous in another, a new analogy arises by taking the remaining terms of the first analogy as extremes, and those of the other as means.

=

Let A B C : D, and B: EF: C, then A: E= F: D.

or a:ef:d; and hence A: EF:D (7).

COR.-If there be any number of analogies such, that two terms not homologous in each are equal to two terms not homologous in the following, a new analogy

will arise by taking the remaining terms of the first as means and those of the last as extremes.

This proposition is commonly called indirect equality (see V. 23).

PROPOSITION XXV. THEOREM.

Ratios that are compounded of equal ratios are equal. This is proved as in proposition F of the other fifth book.

PROPOSITION XXVI. THEOREM.

A compound ratio is equal to the product of its component simple ratios.

A C E

If A: B={C:D, E: F,} then BD F

For, let C: DG: H, and E:F=H:K; and since

G:K=

{G: H, H: K,} (V. Def. 17) = { C : D, E : F,

G g hg

therefore (25) G: K=A:B. But == (Al. Pr.

c

hk

PROPOSITION XXVII. THEOREM.

If four quantities be so related, that when either the first or second increases or diminishes, the third or fourth also respectively increases or diminishes, and also that when the first is a multiple of the second by any terminate number, the third is the same multiple of the fourth; then when the first is a multiple of the second by an interminate number, the third is the same multiple of the fourth.

Let A, B, C, and D, be four quantities such, that when A increases or diminishes so does C, and when B increases or diminishes so does D, and that when A is a multiple of B by any terminate number integral or fractional, C is the same multiple of D; then when A is a multiple of B by an interminate number, C is the same multiple of D; that is, if m be any interminate number, when A=mB, C=mD.

If not, when A = mB, let C = m'D. Then if m" be a terminate number intermediate between m and m', when A becomes A"=m"B, then C becomes C"m"D. Now, if m'm", then m"m; and therefore when the value of A increases from mB to m'B, that of C diminishes from m'D to m'D, which is contrary to hypothesis. Therefore m' cannot be greater than m, and it may similarly be shown that it cannot be less; therefore m' must m, so that when A = mB, C=mD.

H

Schol. This general principle is of easy and extensive application. As an illustration of its application, let it be required to prove that triangles of the same altitude are as their bases. Let EFH, FHG, be the triangles; and first let their bases be commensurable. Let EF contain the common measure M 4 times, and FG contain it 3 times, therefore EF: FG = 4:3 (7). Also, if Ee M, the triangle M HEF contains HEe 4 times, and FGH contains it 3 times (I. 38); hence EFH: FGH = 4:3; and consequently EF: FG = EFH: FGH. Therefore when EF = { FG, EFH = 1 FGH; and hence also, when EF and FG are incommensurable, the same proportion exists (27).

E e

It is obvious that the terminate number m" may always

p

be expressed by some vulgar fraction p and q being integers, and EF or A may be produced to E', so that the measure, which is contained in FG 9 times, may be contained in E'Fp times, and then A"m"B, and also C" =m"D, for EFA", FG = B, E'FH = C", and FGH = D. This will appear very obvious by giving p and q particular values, asp=6, q=5.

In exactly the same manner it may be proved that angles at the centre, or sectors, of a circle, are proportional to their corresponding arcs. The same principle applies also to similar propositions in solid and spherical geometry.

PLANE TRIGONOMETRY.

Plane Trigonometry treats of those relations that subsist between the sides and angles of triangles, by which their numerical values may be computed.

These relations are established by means of certain lines connected with the angles, called trigonometrical lines. When a sufficient number of the parts of a triangle are known, it may be constructed, and then the unknown parts can be measured. This method, however, which is called construction, or the graphic method, would only give a moderate approximation; but, by trigonometrical computation, the values may be found to any required degree of accuracy. The sides are measured by some line of a determinate length, chosen as the unit of measure, as a foot, a yard, a mile, &c.; and the unit of measure of angles is the 90th part of a right angle, or the 360th part of four right angles. As the angles at the centre of a circle are proportional to the arcs subtending them, these arcs may be taken as their measures. The circumference of a circle is accordingly supposed to be divided into 360 equal parts, called degrees; each of these into 60 equal parts, called minutes; each of these into 60 equal parts, called seconds, and so on. If, therefore, a circle be described from the angular point as a centre, the number of degrees, minutes, &c. contained in the arc intercepted by the lines containing the angle, is the measure of that angle. An angle is also sometimes measured by the length of the intercepted arc of a circle, whose centre is the angular point and radius = 1.

DEFINITIONS OF TRIGONOMETRICAL LINES.

1. The complement of an arc is its difference from a quadrant; and that of an angle its difference from a right angle.

2. The supplement of an arc is its defect from a semicircle; and that of an angle is its defect from two right angles.

3. The sine of an arc is a line drawn from one of its extremities, perpendicular to the radius passing through its other extremity.

4. The tangent of an arc is a line touching it at one extremity, and limited by the radius produced through its other extremity.

5. The secant of an arc is that portion of the radius produced, which is intercepted between the extremity of the tangent and the centre.

6. The versed sine of an arc is that portion of the radius intercepted between the sine and the extremity of the arc. 7. The supplemental versed sine, or suversed sine, is the difference between the versed sine and the diameter.

M

F

8. The sine, tangent, &c. of the complement of an arc, are concisely termed the cosine, cotangent, &c. of that arc. These terms, for conciseness, are usually contracted into sin, tan, sec, vers, suvers, cos, cot, cosec, covers, and cosuvers. Let AB be an arc of a circle, AC a quadrant, O the centre; BE, BG, perpendiculars on the radii OA, OC; AF, CH, tangents at A and C; then BC is the complement of AB; and COB that of angle AOB; DCB the supplement of AB, and angle DOB that of AOB ; BE the sine of AB; AF its tangent; OF its secant; AE its versed sine; DE its suversed sine; also BG, CH, OH, the sine, tangent, and secant of BC, or the cosine, cotangent, and cosecant of AB.

D

G

NEE

A

COROLLARIES FROM THE DEFINITIONS.

K

1. The sine of a quadrant, or of a right angle, is equal to the radius.

2. The tangent of half a right angle is equal to the radius. For, if angle AOF were half a right angle, so would F (1.32), and therefore AF would be equal to AO.

3. The sine, tangent, &c. of an arc are equal to those of its supplement.

For BE is the sine of DCB; AF is the tangent of AKL, and therefore of the equal arc DCB; and OF is its secant.