PROPOSITION XXXII. THEOREM.

If two triangles (ABC, CDE) have two sides of the two proportional (AB to BC as CD to DE), and be so placed at an angle, that the homologous sides be parallel, and that the sides not homologous (CB and CD) contain the angle at which they are placed, the remaining sides (AC and CE) shall be in directum.

For since AB and CD are parallel, the alternate angles B and BCD are equal (by Prop. 29, B. 1); and similarly, because CB and ED are parallel, the angles D and BCD A are equal (by same); therefore

B

B is equal to D; and because the sides about these angles are proportional (by Hypoth.), the triangles ABC and CDE shall be equiangular (by Prop. 6, B. 6), therefore the angles ACB and CED are equal; but BCD is equal to CDE; and therefore if DCE be added to both, ACD and DCE together will be equal to CED, EDC, and DCE; therefore ACD and DCE are equal to two right angles (by Prop. 32, B. 1), and therefore AC and CE are in directum (by Prop. 14, B. 1).

In equal circles, angles whether at the centre (AFE and HOL), or at the circumference (AGE and HNL), have the same ratio, that the arches on which they stand have to each other. So also have the sectors (AFE and HOL).

First let the given

angles AFE and HOL
be at the centres, and
divide the arch ACE
into any number of A
equal parts, AC and

N

E

K

CE, and assume HI, IK and KL equal to AC; and draw FC, OI and OK. Therefore, because the arches AC, CE, HI, IK and KL are equal (by Constr.), the angles AFC, CFE, HOI, IOK and KOL are also equal (by Prop. 27. B. 3.); therefore such a submultiple as the arch AC is of ACE, such will the angle AFC be of AFE; and as often as the arch AC is contained in HIKL, so often is the angle AFC contained in HOL; therefore ACE is to HILK, as AFE to HOL (by Def. 5. B. 5.)

If the given angles AGE and HNL be at the circumference, it can be similarly demonstrated that AGE is to HNL, as the arch ACE is to the arch HIKL.

But the sector AFE is also to the sector HOL, as the arch ACE to the arch HIKL.

For letting the preceding construction remain, take any points B and D in AC and CE, and draw BA, BC, DC and DE; and since, if the equal arches AC and CE be taken away from the circle AGEC, the remainders AGC and CGE are equal, the angles ABC and CDE will be equal (by Schol. Prop. 20. B. 3.), and therefore the segments ABC and CDE are similar (by Def. 10. B. 3.); but because the arches ABC and CDE are equal (by Const.), AC, and CE subtending them are also equal (by Schol. Prop. 20. B. 3.), therefore the segments ABC and CDE are equal (by Prop. 24. B. 3.), but the triangles AFC and CFE are equal, because the angles at F, and the sides about them are equal therefore the whole sector AFC is equal to the whole CFE; and it can be similarly shown that these others HOI, IOK and KOL are equal; therefore such a submultiple as the arch AC is of ACE, such will be sector AFC of AFE; but so often as the arch AC is contained in HIKL, so often is the sector AFC contained in HOL; therefore ACE is to HIKL, as the sector AFE to the sector HOL (by Def. 5. B. 5.)

COR. 1.-An angle AFE at the centre, is to four right angles, as the arch on which it stands, is to the whole circumference.

G

For the angle AFE is to a right angle, as the arch on which it stands, to the fourth part of the circumference (by Prop. 33. B. 6.); and therefore to four right angles, as the arch to the whole circumference(by Prop. 30. B. 5.)

H

A

K

COR. 2.-Of unequal circles, the arches which subtend the equal angles, are similar.

For it appears that they have the same ratio to the whole circumference (by preceding Cor.)

COR. 3.-Hence it appears that arches, by which similar segments are contained, are similar.

END OF SIXTH BOOK.

TRIGONOMETRY.

TRIGONOMETRY is of two kinds, Plane and Spherical, it treats of the measurement of heights, distances, &c., by means of triangles. Plane Trigonometry, on which we shall comment, is divided into two parts scilicet, right angled, and oblique, since all triangles must be either rectangular or otherwise. A triangle consists of six parts; id est, three angles and three sides. If any three of these parts be given, provided one be a side, the remaining three can be found; by means of the triangle undergoing the following change; thus if the vertex of either of the acute angles of a right-angled triangle be taken as a centre, and any of the sides as radius, a circle being described, the sides of the triangle receive the following names; if the radius be the longest side: sine, cosine, and radius; and the following, if the radius be either of the other sides: tangent, secant, and radius.

It must be remembered that they retain

D

G

their names with regard to the angle which has been taken as a centre, therefore let ABC be a triangle, and taking AB as radius, and describing the circle BDLgF,

we have BC, the sine of the angle BAC, and AC the cosine of the same angle, but it must be also understood that the arc intercepted between the legs of an angle is the measure of that angle, so that if a circle be divided into 360 degrees, the arc intercepted between the legs of a right angle will be 90 degrees, and so every angle is said to be an angle of as many degrees as are contained in an arc intercepted between its legs. Again, if we take a minor side of the triangle, as radius, and the same angle as centre, and describe the circle CEH, we have the tangent BC of the angle BAC, and AB the secant of the same angle BAC. The circle is usually divided in ENGLAND into 360 degrees, but in FRANCE into 400 degrees; this mode of division has an advantage over that of England, since when degrees, minutes, seconds, and thirds are expressed, which are thus written, 20° 8' 7" 6'", we could also express them in the decimal notation by the very same numbers thus 20.876.

The complement of an angle is what it wants of a right angle, and its sine, tangent, &c., are called the co-sine or sine of the complement, &c. The supplement is what an angle wants of two right angles or a semicircle and the sine, &c. of the supplement is the same as the sine, &c. of that arc or angle.

Now taking the angle BAF of the triangle BAC, or its measure which is either the arc BF or PC, we shall proceed to explain all the lines which can be drawn to the circle DLGF with regard to that angle, that is all the lines used in Trigonometry, as a Cord, derived from Corda, the string of a bow; sine, from sinus, the bosom; versed sine, from sagitta, an arrow; tangent, being the line touching the circle, secant, the line cutting the circle, and the complements of these, which are called co-sine, co-cord, &c. being merely the sine, cord, &c. of the complement of the angle in question.

Thus the sine BC of the angle BAC, or the arc BF, is a line perpendicular to the radius, and meeting an