any line FL, and make the angle LFH=E. On the line FH, take a distance FH equal to AB, the longer side; and with H as a centre, and CD, the shorter side, as radius, describe an arc which will cut FL in two points K, L ; draw the lines HK, HL, and there will be two triangles FHK, FHL,which will satisfy the conditions of the question; that is, two triangles, each of which contains the two given sides and the given included angle.

Scholium. If CD is not of sufficient length to reach the line FL, there will be no triangle which can be formed of the given parts; if it touches FL without cutting it, there will be a right-angled triangle formed, and but one solution.

PROP. XII. PROBLEM.

Through a given point, to draw a tangent to a given circle.

1st. Let the given point B be on the circumference. Draw the radius from C to B; and at the point B draw AB, perpendicular to BC, and AB will be the tangent required (B. II. Prop. 9).

A

Fig. 54.

B

2d. Let the given point B be without the circle.

Join the centre of the given circle and the given point by the line CB; bisect this line in the point O; and with O as a centre, and OC as radius, describe a circumference; it will meet the given circumference in the two points A, D;

draw BA,

D

Fig. 55.

BD, and either of these will be the tangent required; for draw the radii CA, CD, and the angles CAB, CDB, will be right angles, because each is subtended by a semi-circumference (B. II. Prop. 9, and B. II. Prop. 11, Cor. 3).

Cor. The right-angled triangles BDC, BAC, having CB common and CD=CA, are equal (B. I. Prop. 20), and hence BD-BA; therefore two tangents, drawn to the same circle from a point without, are equal.

PROP. XIII. PROBLEM.

To find the common measure of two given straight lines. Let the given lines be AB, CD.

First, suppose CD is contained in AB an exact num

ber of times, that is, without

Fig. 56.

A

C

a remainder; say, for example, four times; then is CD itself the common measure sought; for it is contained in itself once, and in AB, four times. Moreover, any part of CD, as one half, one third, one fourth, &c., is also a common measure of these lines; since it obviously will be contained an exact number of times in CD, and consequently in AB.

B

Secondly, suppose that CD is not contained an exact number of times in AB; but that, after it has been contained in it a certain number of times, (say three, for example,) there is left a remainder EB. Now apply this remainder EB to CD, suppose it is contained twice with a remainder FD; apply this last remainder FD,

to EB, and suppose it is contained four times without a remainder. Now since FD is contained exactly in EB,

it must be contained exactly in CF, which is the double of EB, and consequently in CD; and since it is contained in CD it must be contained in AE, which is three times CD, and consequently in AB; hence FD, being contained exactly in CD and AB, is the common measure of these lines. The given lines are supposed, in the above cases, to have a common measure, and are called commensurable.

Fig. 57.

E

B

It may happen, indeed, that however far we carry the above operation, we shall never find a remainder which is contained an exact number of times in the two lines in this case the lines are called incommensurable ; but it is obvious in this case that the smaller the measuring line is, the smaller the remainder will be; and we may easily suppose the measure taken so small, that the remainder may be neglected without error, and then the lines, to all practical purposes, would be commensurable, that is, have for a common measure this supposed extremely small line.

BOOK IV.

DEFINITIONS.

1. THE ratio of two numbers, is the quotient which arises from dividing one by the other: for example, the ratio of 6 to 2 is 3. Ratio is sometimes expressed by writing the numbers in the form of a fraction; as, or more frequently thus, 6:2, which signifies 6 divided by 2.

Since Geometry relates to magnitudes, and not to numbers, it is necessary to show, that the ratio of magnitudes of the same kind may be expressed by numbers. Let us take, for example, two lines; now to find the ratio between them, or, what is the same thing, the number of times that one is contained in the other, we may suppose them both divided into equal parts, (for example, inches,) and let us say that one contains sixty-two inches, and the other thirty-seven inches; now the number of times that one of the lines contains the other, will evidently be the same as the number of times which sixty-two contains thirty-seven; that is, the ratio of the lines would be expressed by, or 62 divided by 37. Hence the same rules which relate to the ratio of numbers will apply to the ratio of magnitudes.

2. Four numbers are in proportion, when the ratio, or quotient, of the first and second, is the same as the ratio of the third and fourth: for example, 12, 6, 8, 4 are in pro

portion, because 12 divided by 6, is equal to 8 divided by 4.

To indicate a proportion, numbers are written thus ; 12:6 :: 8:4, which is read, 12 is to 6, as 8 is to 4.

3. The numbers which make a proportion are called the terms of the proportion; the first and last terms are called the extremes, the second and third the means.

4. The first and third terms are called antecedents, the second and fourth consequents.

5. Three numbers are in proportion when the ratio of the first and second is equal to that of the second and third: for example, 12 : 6 :: 6 : 3; in this case the middle term is said to be a mean proportional to the other two.

6. When two numbers are multiplied by the same number the results are called equimultiples of the two numbers: as, if 8 and 6 are both multiplied by 4, then 32 and 24 are equimultiples of 8 and 6.

7. If we have several numbers multiplied by the same number, as 6.4, 8.4, 10.4, we may express the sum of all these products thus, (6+8+10).4. (Explanation of Signs, 6).

Prop. 1. When four numbers are in proportion, the product of the means is equal to the product of the ex

tremes;

as,

6:4: : 15:10, hence 10.6=15.4.

Prop. 2. If the product of two numbers is equal to the product of two other numbers, the four may be arranged in a proportion ;

as,

8.3-12.2, hence 8: 12::2:3.

Scholium. Remark that the test of a proportion, in every case, is, that the product of the extremes is equal to that of the means; hence, when we have four quantities to arrange in a proportion, the terms may be placed