cutting the circle in C, and the fixed line in D, so that AC shall be equal to CD. Let the fixed line meet the o in F; bisect the arc AB in E, draw EC 1 to AF,


85. Describe a circle which shall touch a given straight line at a given point, and pass through another given point not in the line.

86. Construct a triangle, having given the base, the vertical angle, and the median drawn from the vertical angle.

87. ABC is a given straight line: find a point P such that each of the angles APB, BPC may be equal to a given angle.

88. Find the point inside a given triangle at which the sides subtend equal angles.

89. About a given circle to describe a triangle equiangular to a given triangle. On two sides describe segments of Os containing an L = of 2 rt. Zs, etc.

90. If the escribed circles of the triangle ABC (272) touch BC, CA, AB externally in D, E, F respectively, prove that BD = EA, CD = AF, CE = BF.

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277. Four quantities are said to be in proportion when the ratio of the first to the second is equal to the ratio of the third to the fourth.

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then, A, B, C, D are said to be in proportion. The proportion is written

A :B= C:D,

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NOTE.-The first form is preferable, and the one most generally used in the higher mathematics; but the second form is the more usual one in elementary works.

Let a and b denote the numerical measures of A and B (229), and c and d the numerical measures of C and D. Then, since the ratio of two quantities is the same as the ratio of their numerical measures (230),

..A:B = a:b, and

C:D=c:d. Hence, if four quantities A, B, C, D, are in proportion, their numerical measures a, b, c, d, are in proportion; that is, a:=c:d.

Conversely, if the numerical measures of four quantities A,B,C,D, are in proportion, the quantities themselves are in proportion; that is,


when A and B are quantities of one kind, and C and Dare quantities of one kind, though the latter kind may be different from the former.

That is, all four quantities may be of the same kind, as, for instance, four straight lines, four surfaces, four angles, and so on; but the quantities in each pair must be of the same kind.

The magnitudes we meet with in Geometry are more often incommensurable (232) than commensurable.

The preceding reasoning does not apply directly to the .case in which two quantities are incommensurable, but it may be extended to this case.

278. To find the greatest common measure of two quantities.

Let there be two quantities, as, for instance, the two straight lines


Éd Apply the smaller CD to the greater AB, as many times as possible, suppose twice, with a remainder EB.

Apply the remainder EB to CD as many times as possible, suppose once, with a remainder FD.

Apply the second remainder FD to EB as many times as possible, suppose once, with a remainder GB.

Apply the third remainder GB to FD as many times as possible.

This process will terminate only if a remainder is found which is contained an exact number of times in the preceding one; and if it so terminates the two given lines are commensurable, and the last remainder will be their greatest common measure.

Suppose, for example, that GB is contained exactly twice in FD.

Then GB is contained exactly in CF, and therefore exactly in CD, and therefore exactly in AE, and therefore exactly in AB.

Therefore GB is a common measure (231) of AB and CD.

And, since every common measure of AB and CD must divide AE, it must divide EB or CF, and therefore FD, and therefore also GB. Hence, the common measure cannot be greater than GB. Therefore GB is the greatest common measure of AB and CD.

By regarding GB as the measuring unit, the values of the preceding remainders are easily found, and finally, those of the given lines, from which their numerical ratio is ob tained.

Thus, FD = 2GB; EB = FD + GB = 3GB;

CD= EB + FD = 2FD + GB = 5GB;

AB = 2CD + EB = 10GB + 3GB = 13GB. Therefore the given lines AB and CD are numerically expressed in terms of the unit GB by the numbers 13 and 5; and their ratio is 13

279. When the quantities are incommensurable the above process never terminates. However far the operation is continued, we never find a remainder which is contained an exact number of times in the preceding one. But as there is no limit to the number of parts into which AB and CD may be divided, we may obtain a remainder as small as we please, one that is less than any assignable quantity.

Hence, although no finite numerator and denominator, however large, can exactly express the ratio of two incommensurable quantities, yet by properly increasing both numerator and denominator we may obtain a ratio as nearly equal as we please to the required ratio, i.e., we may obtain two numbers whose ratio will express the ratio of two incommensurable quantities to any required degree of accuracy (232).

NOTE.—We therefore conclude that ratio in Geometry may be treated in the same way as ratio in Arithmetic and Algebra. Indeed, the algebraic treatment is the easiest and the simplest. Euclid's treatment of ratio and proportion is now practically disregarded. While his reasoning is exquisite and rigorous, it is remote from our practical notions on the subject. For these reasons, only simple algebraic proofs of the propositions in proportion will be given in this work. The student will perceive that the propositions do not introduce new ideas, but merely supply proofs, based on the geometric definition of proportion, of results already familiar in the study of Algebra.

280. Let us now consider the numerical proportion (277), which may be written in either of the three forms:

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The four terms of the two equal ratios (277) are called the terms of the proportion. The first and fourth terms are called the extremes, and the second and third, the means. Thus, in the above proportion, a and d are the extremes, and b and c the means.

The first and third terms are called the antecedents, and the second and fourth the consequents. Thus, a and c are the antecedents, and b and d the consequents.

The fourth term is called a fourth proportional to the other three. Thus, in the above proportion, d is a fourth proportional to a, b, and c.

In the proportion a:b=b:c, c is a third proportional to a and b, and b is a mean proportional between a and c.

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