III. Part is a number of a number; the leffer of the greater, when the leffer measureth the greater. Every part is denominated from the number, by which it measures the number whereof it is a part; as 4 is called the third part of 12, because it measures 12 by 3. IV. But when the leffer number does not measure the greater, then the leffer is call'd, not a part, but parts of the greater. All parts whatsoever are denominated from these two numbers, by which the greatest common measure of the two numbers measures each of them; as 10 is faid to be two thirds of the number 15; because the greatest common measure, which is 5, measures 10 by 2, and 15 by 3. V, A multiple is a greater number compared with a leffer, when the leffer measures the greater. VI. An even number is that which may be divided into two equal parts. VII. But an odd number is that which cannot be divided into two equal parts; or that which differeth from an even number by unity. VIII. A number evenly even, is that which an even number measureth by an even number. IX. But a number evenly odd, is that which an even number measureth by an odd number. X. A number only oddly odd, is that which an odd number measureth by an odd number. XI. XI. A prime (or firft) number is that which is meafured only by unity. XII. Numbers prime the one to the other, are fuch as only unity doth ineafure, being their common measure. XIII. A compofed number is that which fome certain number measureth. XIV. Numbers compofed the one to the other, are thofe, which fome number, being a common measure to them both, doth measure. In this, and the preceding definition, unity is not a number. XV. One number is faid to multiply another when the number multiplied is fo often added to it self, as there are units in the number multiplying, and another number is produced, Hence in every multiplication unity is to the multiplier, as the multiplicand is to to the product. Obf. That many times, when any numbers are to be multiplied (as A into B) the conjunction of the letters denotes the product: So A BAX B, and CDE = CxD × E. XVI. When two numbers multiplying themfelves produce another, the number produced is called a plane number; and the numbers which multipled one another, are called the fides of it: So 2 (C) × 3 (D) = 6 CD is a plane number. XVII. But when three numbers multiplying one another produce any number, the number produced is termed a folid number; and the numbers multiplying one another, are called the fides thereof: So 2 (C) X 3 (D) × 5 (E) = 30 =30= CDE is a folid number. XVIII. A square number is that which is equally equal; or, which is contained under two equal numbers. Let A be the fide of a fquare; the Square is thus noted, AA, or Aq. XIX. A Cube is that number which is equally equal equally; or which is contained under three equal numbers. Let A be the fide of a Cube; the Cube is thus noted, AAA, or Ac. In this definition, and the three foregoing, unity is number. XX. Numbers are proportional, when the first is the fame multiple of the fecond, as the third is of the fourth; or, the fame part; or, when a part of the first number measures the fecond, and the fame part of the third third measures the fourth, equally: and vice versa. So A: B::C: D. that is, 39 9: : 5:15. XXI. Like plane, and folid numbers, are those which have their fides proportional: Namely, not all the fides, but fome. XXII. A perfect Number is that which is equal to its own parts. As 6, and 28. But a number that is less than it's parts is called an Abounding number, and one which is greater, a Diminutive: fo 12 is an abounding, 15 a diminutive number. XXIII. One number is faid to measure another, by a third number, which when it either multiplies, or is multiplied by the measuring number, produces the number measured. In Divifion, unity is to the quotient, as the divifor is to the dividend. Note, that a number placed under another A with a line between them, fignifies divifion: Só—— A diB Those two numbers are called the Terms or Roots of a Proportion, than which leffer cannot be found in the fame proportion. T Poftulates, or Petitions. Hat numbers equal or multiple to any number may be taken at pleasure. 2. That a number greater than any other whatsoever may be taken. 3. That Addition, Subtraction, Multiplication, Divifion, and the Extractions of Roots or fides of fquare and cube numbers, be alfo granted as poffible. 1. W Axioms. Hatfoever agrees with one of many equal numbers, agrees likewife with the reit. 2. Thofe parts that are the fame to the fame part, or parts, are the fame among themselves. 3. Numbers that are the fame parts of equal numbers, or of the fame number, are equal among themfelves. G 4. Those a 11. ax. 7. b 12.ax. 7. cg.ax. I. 4. Thofe numbers of which the fame number or equal numbers, are the fame parts, are equal amongst themselves. 5. Unity measures every number by the units that are in it, that is, by the fame number. 6. Every number measures it fself by unity. 7. If one number multiplying another, produces a third, the multiplier fhall meature the product by the multiplied; and the multiplied fhall measure the fame by the multipler. Hence, No prime number is either a plane, folid, Square, or cube number. 8. If one number measures another, that number by which it measureth shall measure the fame by the units that are in the number measuring, that is, by the number it felf that measures. حرية 9. If a number measuring another, multiply that by which it meafureth, or be multiplied by it, it produceth the number which it measureth. 10 How many numbers foever any number measureth, it likewise measureth the numbers compofed of them. 11. If a number measures any number, it also meafureth every number which the faid number measureth. 12. A number that measures the whole and a part taken away, doth alfo measure the refidue. PROP. I. A..... E. . G. BS 5 3 C...F.. D. ૐ I H-- Ι 2 Tavo unequal numbers AB, EB from CD, &c.) by an alternate fubtraction, and the If you deny it, let AB, CD, have a common measure, namely the number H; therefore H measuring CD, doth (a) alío measure AE; and (b) confequently the remainder EB; (a) therefore it likewife measures CF, and (b) fo the remainder FD; (a) therefore it also measures EG. But it measured the whole EB, and (b) therefore it must meafure that which remaineth GB, that is, a number meafures unity. () Which is abfurd. PROP. PROP. II. Two numbers AB, CD being given, not prime the one to the other, to find out their greatest common measure FD. Take the leffer number CD from the greater AB as often as you can. If nothing remains, (a) it is manifeft that CD is the greatest common measure. But if there remains fomething (as EB) then take it out of CD, and the refidue FD out of EB, and fo forward till fome number (FD) measure the faid EB, (b) for this will be, before you come to unity; FD fhall be the greatest common measure. For FD (c) measures EB, and (d) therefore alfo CF; and (e) confequently the whole CD; (d) therefore likewise AE; and fo measures the whole AB. Wherefore it is evident that FD is a common meafure. If you deny it to be the greatest, let there be a greater (G); then whereas G measureth CD, it (d) must likewife measure AE, (e) and the refidue EB, (d) as alfo CF, (e) and by confequence the refidue FD, (g) the greaterthe less. (b) Which is abfurd. Coroll. Hence, a number that measures two numbers, does also measure their greatest common measure. PROP. III. Three numbers being given, A, B, C, not prime one to the other, to find out their greatest common measure E. D.... 4• C ......6 E..2 If D does not mea Find out D, the greatest common measure of the two numbers A, B. If D measures C the third, it is F... clear that D is the greatest common measure of all the three numbers. fure C, at leaft D and C will be composed the one to the other, by the Coroll. of the Propofition preceding. Therefore let E be the greatest common measure of the faid numbers D and C, and it shall be the number which was required. a 6.ax. 7. b 1.7. c cenftr. d11.ax. 7. e 12.ax. 7. d 11.ax. 7. d11. e 12.ax. 7. g Juppos. |