1871. III. 32. The opposite sides of a quadrilateral inscribed in a circle are produced to meet in P, Q, and about the four triangles thus formed circles are described: prove that the tangents to these circles at P and Q form a quadrilateral equal in all respects to the original, and that the line joining the centres of the circles, about the two quadrilaterals, bisects PQ. 1872. iv. 5. A triangle is inscribed in a given circle so as to have its centre of perpendiculars at a given point prove that the middle points of its sides lie on a fixed circle. 1. 47 If CE, BD be the squares described upon the side AC, and the hypotenuse AB, and if EB, CD intersect in F, prove that AF bisects the angle EFD. III. 22. Two circles intersect in A, B: PAP', QAQ' are drawn equally inclined to AB to meet the circles in P, P', Q, Q': prove that PP' is equal to QQ'. IV. 4. Having given an angular point of a triangle, the circumscribed circle, and the centre of the inscribed circle, construct the triangle. BOOK V. SECTION 1. On Multiples and Equimultiples. DEF. I. A GREATER magnitude is a Multiple of a less magnitude, when the greater contains the less an exact number of times. DEF. II. A LESS magnitude is a Sub-multiple of a greater magnitude, when the less is contained an exact number of times in the greater. These definitions are applicable not merely to Geometrical magnitudes, such as Lines, Angles, and Triangles; but also to such as are included in the ordinary sense of the word Magnitude, that is, anything which is made up of parts like itself, such as a Distance, a Weight, or a Sum of Money. POSTULATE. Any one magnitude being given, let it be granted that any number of other magnitudes may be found, each of which is equal to the first. METHOD OF NOTATION. Let A represent a magnitude, not as one of the letters used in Algebra to represent the measure of a magnitude, but let A stand for the magnitude itself. Thus, if we regard A as representing a weight, we mean, not the number of pounds contained in the weight, but the weight itself. Let the words A, B together represent the magnitude obtained by putting the magnitude B to the magnitude A. Let A, A together be abbreviated into 24, A, A, A together and so on. ............3A, Let A, A......repeated m times be denoted by mA, m standing for a whole number. Let mA, mA......repeated n times be denoted by nmA, where nm stands for the arithmetical product of the whole numbers n and m. Let (m+n) A stand for the magnitude obtained by putting nA to mA, m and n standing for whole numbers. These, and these only, are the symbols by which we propose to shorten and simplify the proofs of this Book: capital letters standing, in all cases, for magnitudes; and small letters standing for whole numbers. SCALES OF MULTIPLES. By taking a number of magnitudes each equal to A, and putting two, three, four......of them together, we obtain a set of magnitudes, depending upon A, and all known when A is known; namely, A, 2A, 3A, 4A, 5A................................and so on ; each being obtained by putting A to the preceding one. This we call the SCALE OF MULTIPLES of A. If m be a whole number, mA and mB are called Equimultiples of A and B, or, the same multiples of A and B respectively. AXIOMS. 1. Equimultiples of the same, or of equal magnitudes, are equal to one another. 2. Those magnitudes, of which the same, or equal, magnitudes are equimultiples, are equal to one another. 3. A multiple of a greater magnitude is greater than the same multiple of a less. 4. That magnitude, of which a multiple is greater than the same multiple of another, is greater than that other magnitude. NOTE 1. If A and B be two commensurable magnitudes, it is easy to show that there is some multiple of A, which is equal to some multiple of B. For let M be a common measure of A and B; then the scale of multiples of M is M, 2M, 3M,..................... Now one of the multiples in this scale, suppose pM, is equal to A, .......suppose qM,............................................B. and one PROPOSITION I. (Eucl. v. 1.) V. Ax. 1. I. Ax. 1. If any number of magnitudes be equimultiples of as many, each of each; whatever multiple any one of them is of its submultiple, the same multiple must all the first magnitudes, taken together, be of all the other, taken together. Let A be the same multiple of C that B is of D. Then must A, B together be the same multiple of C, D together that A is of C. Let A = = C, C, C............repeated m times. = .. A, B together C, D; C, D; C, D;......repeated m times. .. A, B together is the same multiple of C, D together that A is of C. E. D. PROPOSITION II. (Eucl. v. 2.) If the first be the same multiple of the second that the third is of the fourth, and the fifth the same multiple of the second that the sixth is of the fourth; the first together with the fifth must be the same multiple of the second, that the third together with the sixth is of the fourth. Let A, B, C, D, E, F be six magnitudes, such that and A is the same multiple of B, that C is of D, and C, F together = D, D, D,............repeated m+n times. .. A, E together is the same multiple of B, that C, F together is of D. PROPOSITION III. (Eucl. v. 3.) Q. E. D. If the first be the same multiple of the second that the third is of the fourth; and if of the first and third there be taken equimultiples, these must be equimultiples, the one of the second, and the other of the fourth. Let A be the same multiple of B that C is of D ; and let E and F be taken equimultiples of A and C. Then must E and F be equimultiples of B and D. For let A then C Again, let E then = = = B, B,........ D, D,...... A, A,..... FC, C,. .. EmB, mB, and FmD, mD,... .repeated m times = mB; ..repeated m times=mD. ..repeated n times; ..repeated n times. .repeated n times=nmB; ...repeated n times=nmD. .. E is the same multiple of B that F is of D. Q. E. D. |