PROPOSITION X. If two great circles intersect, their points of intersection will be the poles of the great circle which passes through their poles. For (in Fig. 5), take the plane of the paper for the plane of one circle, and YPX for any other, so that X and Y are the points of intersection of the two circles; through O draw a plane APB perpendicular to XY, and .. perpendicular to both the planes XAB, and XPY, and.. the plane APB will contain the lines drawn perpendicular to those planes, and therefore will contain the poles of the two given circles. Hence APB is the great circle joining the poles of the circles XAY and XPY; but YOX is perpendicular to the plane APB, .. X and Y are the poles of the circle APB; i. e., are the poles of the great circle which joins the poles of the two given circles. PROPOSITION XI. If ABC is a given triangle, and A'B'C', is its polar triangle, then is ABC the polar triangle of A'B'C'. For since C' is the pole of AB, and B' is the pole of AC,.. (by last Prop.) the point of intersection A of AB and AC is the pole of the great circle joining B'C', i.e., A is the pole of B'C', similarly B is the pole of C'A', and C the pole of A'B'. Q. E. D. PROPOSITION XII. If ABC is a triangle, and A'B'C' its polar triangle, B B P Fig. 10. For, produce AB, AC to meet B'C' in P and Q. Then because AP and AQ are quadrants, PQ is the arc that measures the angle A (Cor. 4, Prop. V.); now B·C′+PQ=B'P + C'Q + QP = B'Q+ C'P. But since B' is the pole of AC, .. BQ is a quadrant. Similarly CP is a quadrant, and the two together are a semicircle, .. B'C', together with the arc on the great circle which measures A, equals the semi-circumference of a great circle. COR. 1.—If for these arcs we substitute the angles they measure, we may state the proposition as follows: ABC 2 right angles. Similarly B+C'A' 2 right angles, C+A'B' 2 right angles. COR. 2.-And since ABC is the polar triangle of A'B'C', we have ABC2 right angles. COR. 3.-Hence, the sum of the angles of any triangle, together with the sides of the polar triangle, = six right angles. But the sides of the polar triangle must have some magnitude, and must be less than four right angles (Prop. VII.) Hence the three angles of a triangle must be less than six, and greater than two right angles. It is plain, since the three angles of a spherical triangle are greater than two right angles, and less than six right angles, that a spherical triangle may have one, two, or even three of its angles right angles. DEF. XIII.—A right-angled spherical triangle is one which has one or more right angles. DEF. XIV.-A quadrantal triangle is one which has at the least one side a right angle, i.e., the quadrant of a great circle. PROPOSITION XIII. If ABC is a right-angled triangle, having a right angle C, and A'B'C' is its polar triangle, then A'B'C' is a quadrantal triangle, having the side A'B' a quadrant. For by the last proposition (Cor. 1)— C + A'B' = two right angles. Now, C is a right angle, .. A'B' is a right angle, i. e., is a quadrant. Q. E. D. COR.-Hence, if all three angles, A'B'C, are right angles, the sides of the polar triangle are all right angles. For if two sides of a triangle are right angles, the third side measures the opposite angle; .. if the third side is also a right angle, all the angles are right angles. Hence in the polar triangle the sides and angles are all right angles. And since the angles of the polar triangle are each right angles, the sides of ABC will be right angles (last Prop., Cor. 2). Hence, if all the angles of any triangle are right angles, the sides are right angles (i.e., quadrants) also. SERIES AND LOGARITHMS. ON SERIES. 1. The Principle of the Permanence of Equivalent Forms. It was stated in page 161, that “the processes of Algebra are, for the most part, only processes of Arithmetic, extended and rendered more comprehensive by the aid of a new set of symbols, taken in combination with the well-known symbols of Arithmetic;” and in the explanations following, it is made to appear that Algebra is a generalization of Arithmetic—that whereas 2, 5 ... represent certain special numbers, a, b ... represent any numbers. This is professedly an elementary view of the case ; and, as an elementary view, is quite sufficient. But when the nature of the generalization is more closely considered, it appears that, in what is commonly called Algebra, there are really two distinct, though closely connected sciences, which may be called respectively Arithmetical Algebra, and Symbolical Algebra. In Arithmetical Algebra, “the symbols represent numbers, whether abstract or concrete, whole or fractional, and the operations to which they are subject are assumed to be identical, in meaning and extent, with the operations of the same name in common arithmetic. The only distinction between the two sciences consists in the substitution of general symbols for digital numbers.” Thus, in arithmetic, it is impossible to subtract 7 from 5: so that 5 -7 is impossible; and hence in arithmetical Algebra, when we write a -6, we do so with the tacit assumption that a 7 b. If we generalize a step farther than this, and allow ourselves to write a ~ 6 for all values of a and b, then it is clear that the negative sign has a more extended meaning than that of mere subtraction; and it remains for us to ascertain what this more extended meaning is. The science which concerns itself with this second generalization is called Symbolical Algebra. Thus, then, we have, in all three sciences,— (1). Arithmetic, in which the symbols employed are particular in form, and particular in value. (2). Arithmetical Algebra, in which the symbols employed are general in form, but particular in value. (3). Symbolical Algebra, in which the symbols employed are general in form, and also general in value. Thus, as the second of these sciences is a generalization of the first, so the third is a generalization of the second. The principle in accordance with which this second generalization is conducted called that of “The permanence of equivalent forms." The principle may be stated as follows : “Whatever algebraical forms are equivalent, when the symbols are general in form, but specific in value, will be equivalent likewise when the symbols are general in value, as well as in form.” For the full exposition of relations between these two sciences, the advanced reader is referred to a “Treatise on Algebra," by George Peacock, D.D., to whom is due the detection of the coexistence of these two sciences in that which is generally treated as one science-Algebra. We shall have several occasions to make use of the principles above enunciated in the course of the following pages. As an example of their application, we will reconsider the Theory of Indices already treated in pp. 191, 192, 193, : 2. On the Theory of Indices. We have already seen that a” signifies a × a × a ×a, &c., to m factors, when, of course, m must be a whole number. In like manner a" signifies a × a × a, &c., to n factors. Hence am xanaxaxa.... to (mn) factors, and therefore am X anam + n ̧ This is a result in Arithmetical Algebra. It is perfectly general in form, but it is particular in value; for m and n are, by the definition of am, limited to being positive whole numbers. If we suppose m and n to have negative or fractional values, this involves a generalization of our original definition, and the question arises what meaning we must assign to such expressions as an a. To answer it, we proceed in the following manner:-By assuming m and ʼn general in value as well as in form, we enter the domains of Symbolical Algebra; hence, by the principle of the permanence of equivalent forms, under all circumstances P .. am × an × a” = am ÷n × a2 = am +n+r ; and so on for any number of terms. Hence i.e., a must (in accordance with our general principle) signify the 4th root of the pth power of a. Hence, we see that in assigning the meaning √ a to a2, we are doing so not arbitrarily, but in accordance with a principle which lies at the foundation of Algebra. 3. On Impossible Expressions. -- In symbolical a and v a are Again, we know that va2= a. In like manner if we were to have — a2, this is = a2 × (— 1), and .•. √ — a2 = a√ 1. The expression VI is frequently spoken of as an "impossible quantity," an "imaginary expression," and so on, since 1 cannot be produced by multiplying either + 1 by + 1, or 1 by 1. In reality, however, V-1 is as possible or as impossible as - 1; for in arithmetical Algebra a and a are only admissible on the supposition that a is positive. Algebra this restriction is removed, and therefore in that both admissible. Hence in future investigations we shall make use of V 1 just as freely as va, whenever it may suit our purpose, quite undeterred by the circumstance of its so-called impossibility. Of course there are many differences between the symbols - 1 and I; for instance, the interpretation of the former is a much simpler matter than the interpretation of the latter, and in some cases a — b belongs to arithmetical Algebra; but a + by 1 never does. We cannot enter further into the matter now; 2 X - 2 what we have said will be enough to explain that we are justified in introducing into our calculations expressions which are called "imaginary,” or “impossible.” 4. To prove that if A. + Apk + A2x2 + ... + Anando + az x + Q2 22 +.... + an In, for all values of x. Then A, = au, A2 = , A, = 029 An= an. 'or since A, + 423 +...+ An X" = do + ax + + anzn are equivalent for ALL values of x; they are equivalent when x=0. ::. Ao = = do, and :. A& + A,2" t. + Anx = ax + aan te + An xn ; -, A, + A, u to. + Anan-1= a, + 22 + + an an – 1 for all values of x; and hence when x = 0, -, A2 = an, and and so on. Hence A, = Qg ... and An = an. This is called the principle of Indeterminate Coefficients. It will be seen that in the case supposed, where the number of terms in each series is finite, the proof is quite rigid. If each series were infinite the proof would not then be conclusive; and, accordingly, we shall refrain from using this principle except in cases where no objection can be raised to its use. Such as the following : 1-2x + 3x2 (1). Toresolve (x - 1) (x - 2) (x – 3) into partial fractions. 1 Assume 2x +322 A, A, A, (— 1) (2 — 2) (7—3) + -1 -3. :: 3x2 – 2x+1= A(x-2)(x - 3) + Ag (2 - 3)(- 1) + A3(2-1) (x - 2) = *P(A, + A2 + A3) — 2(5A, + 4A, + 3A3) + 6A, 3A, + 2Ag. This being true for all values of x, we have A, + A, + A3=3. .. 24, + Aq=-7. 2A, :: A, = 1. Ag=11. 1 9 11 (1-1) (0-2) (-3) -1 ( N.B. A fraction written in the above form is said to be resolved into its partial fractions. (2). Resolve (x + 1) (** +3) into its partial fractions. 1 A Mr+N + :. *—1=x(A = M) + x(M + N) + 3A + N M+N=1 3A +N=-1 A= + 2 -- 2 X-3 X-1 72 . |