PROP. 60. If A, B, C, D be four harmonic points, A and C being conjugate, and if O be the middle point of AC, then OC is a mean proportional between OB and OD. 63. If K, L, M, P be four straight lines in proportion, if the lengths of L and M be fixed, if the length of K can be made smaller than that of any line however small, to show that the length of P can be made greater than that of any PROP. 64. If A, B, C, D are magnitudes of the same kind, and if A : B=C : D, then 65. If A, B, C, D are magnitudes of the same kind, and if A: B=C : D, then the sum of the greatest and least of the four magnitudes is greater than that of A : B : : C : D The ratio of A to B is the same as the ratio of C to D. [A, B] ~ The relative multiple scale of A, B. (Art. 33.) Is the same as. (Art. 33.) [A, B]≈ [C, D] The relative multiple scale of A, B is the same as the relative multiple scale SECTION I. PROPOSITIONS 1-8. ON MAGNITUDES AND THEIR MULTIPLES. Art. 1. Number. In this book except where otherwise stated the word Number will be used as an abbreviation for Positive Whole Number. Notation for Number. A Number will always be denoted by a small letter. Art. 2. Notation for Magnitude. A Magnitude will be denoted throughout this book by a capital letter*. Art. 3. Def. 1. MULTIPLE. One magnitude is said to be a multiple of another magnitude, when the former contains the latter an exact number of times. It will be in agreement with the above nomenclature, when A is equal to B, to say that A is the first multiple of B; and to call the rth multiple of B and the rth multiple of C the same multiples of B and C. Art. 4. It is necessary to prove certain propositions regarding magnitudes and multiples of magnitudes before entering upon the discussion of the relations between magnitudes. * A point will also be denoted by a capital letter, but this will not lead to any difficulty. H. E. 1 ENUNCIATION. Art. 5. PROPOSITION I.* (Euc. v. 1.) To prove that r(A+B)=rA+rB. Construct the following diagram. Draw a rectangle. Draw one line parallel to one pair of sides, dividing it into two compartments. Draw (r− 1) lines parallel to the other pair of sides, dividing each of the two compartments into r compartments. In or upon each one of the upper row of compartments place the magnitude A; and in or upon each one of the lower compartments place the magnitude B. Then, adding together the two magnitudes in any column the result is A + B, and as there are r columns, the sum of all the magnitudes on the whole rectangle is r(A+B). Again, adding together the magnitudes in the upper row the result is rA, and the sum of the magnitudes in the lower row is rB. Hence the sum of all the magnitudes is rA +rB. But the sum of all the magnitudes is independent of the order in which they are added. .. r(A+B)=rA +rB. Art. 6. EXAMPLE 1. By repeated application of Proposition I. prove that r (A + B + ... + K) = rA + rB + + rK. Art. 7. PROPOSITION II.* (Euc. v. 2.) ENUNCIATION. To prove that (a + b) R = aR+bR. Take any rectangle. Draw (a + b − 1) straight lines parallel to one pair of sides, thus dividing it into (a+b) compartments. * See Note 1. Then the sum of all the magnitudes is (a + b) R. Again, separating the magnitudes into two groups, consisting respectively of the magnitudes in the first a compartments, and the magnitudes in the remaining b compartments, the sum of the magnitudes in the first group is aR, and the sum of those in the second group is bR. Hence the sum of all the magnitudes is aR + bR. But this sum was shown above to be (a + b) R. .. (a + b) RaR+bR. 3. If A and B are both multiples of G, prove that A + B is a multiple of G. ENUNCIATION. If A > B, then r (A – B) = rA − rB. Since let But |