quall equally'; or, which is conteined under three equall numbers. Let A be the fide of a Cube; the Cube is thus noted,AAA, or Ac. In this definition, and the three foregoing, unitie is a number. XX.Numbers are proportionall,when the first is as multiplex 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 measures the fourth, equally and on the contrary. So A.B:: C. D.that is, 3.9: 5.15. XXI, Like Plane, and folid numbers are they which have their fides proportionall: Namely, not all the fides, but fame. XXII.A Perfect number is that which is equall to its own parts. As 6. 28.But a number that is leffe then it's j parts is called an Abounding number; and a greater a Dimiautive: fo 12 is an abounding, 15 a diminutive number. XXIII.One number is faid to measure another,by that number, which, when it multiplies, or is multiplied by it, it produceth. In Divifion,a unitie is to the quotient as the divifor is to the dividend. Note, that a number placed under another with a line between them, fignifies divifion : So AA divided by B,& CA Cx A divided by B. B Two numbers are called Termes or Roots of Proportion, leffer then which cannot be found in the fame proportion. Poftulates, or Petitions. 1. That numbers equall or manifold to any number may be taken at pleasure. 2. That a greater number wharfoever. number may be taken then any 3. That Addition, Subtraction, Multiplication, Di Divifion, and the Extractions of roots or fides of fquare and cube numbers,be alfo granted as poffible. Axiomes. IW Hatfoever agrees with one of many equall numbers, agrees likewife with the rest. 2. Those parts that are the fame to the fame part, or parts, are the fame amongst themselves. 3. Numbers that are the fame parts of equall numbers,or of the fame number, are equall amongst themselves. 4. Those numbers,of whom the fame number, or equall numbers, are the fame parts, are equall amongst themselves. 5. Unitie measures every number by the unities that are in it; that is,by the fame number. 6. Every number measures it self by a unitie. 7. If one number, multiplying another, produce a third, the multiplier fhall measure the product by the multiplied; and the multiplied fhall measure the fame by the multiplier. Hence, No prime number is either a plane, folid, Square, or cube number. 8. If one number measure another, that number by which it measureth fhall measure the fame by the unities that are in the number measuring, that is, by the number it felf that measures. 9. If a number measuring another, multiply that by which it measureth, or be multiplied by it,it pro duceth the number which it measureth. 10. How many numbers fo ever any number meafureth, it likewife measures the number composed of them. 11. If a number meafure any number, it alfo measureth every number which the faid number measureth. 12. A number that measures the whole & a part taken PRO PROPOSITION I. A...... E.. G.B 853 52 3 given, the beffer CD be the greater AB (and the refidue EB from CD,&c.) by an alternate fubftraction, and the number remaining do never measure the precedent,till the unitie GB be taken; then are the numbers which were given AB,CD, prime the one to the other. If you deny it,let AB, CD have a common meafure, namely the number H. Therefore H measuring CD, dotha alfo measure AE; and confequently the a 11.42 9. remainder EB; therefore it likewife meafures CF, and fo the remainder FD; a wherefore it alfo meafures EG. But it meafured the whole EB,and therefore it must measure that which remaineth GB,a unitie, it felf being a number. Which is Absurd. their greatest com- b 18.ax..! Take the leffer number CD from the greater AB as often as you can. If nothing remains, it is mani a 6.x.. feft 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 (for this will be,before you come to a unitie.) FD b 1.7 thall be the greatest common measure. C d For FD measures EB, and therefore alfo CF; ande confequently the whole CD; therefore likewife AE; and fo measures the whole AB. Wherefore C confr. duax.f. € 13.4x.7. d11.4x,7. € 22.4x.7. B fuppof 19.4x 1. a conftr. b11.ax.7. C cor. 1.7. d fuppof. it is evident that FD is a common measure. If you deny it to be the greateft,let there be a greater (G;) then whereas G measureth CD, it must likewife measure AE, e & the refidue EB,d as also CF, e and by confequence the refidue FD, g the greater the leffe.Which is abfurd. Coroll. Hence, A number that measures two numbers, does also measure their greatest common measure. A............ 12 PROP. III. Three numbers being given, A, B, C, not prime to one another, to find D....4 out their greatest common measure 8 C...... 6 E.. 2 E. Find out D the greatest.common measure of the two numbers A,B.If D measures C the third, it is clear that D is the greatest common measure of all the three numbers. If D does not measure C, at , least D and C will be compofed the one to the other,by the Coroll.of the Prop. preceding. Therefore let E be the greatest common measure of the faid numbers D and C, and it appears to be the number required. For E a measures C and D, and D measures A and B; therefore b E measures each of the numbers A,B,C:neither fhall any greater (F) measure them, for if you affirm that, then F measuring A and B, does likewise measure D their greatest common measure; and in like manner,F measuring D and C, does alfo measure Ec their greatest common meafure, d the greater the leffe. e Which is abfurd, Coroll. Hence, A number that measures three numbers, does alfo measure their greatest common meafure. PROP. Every leffe number A is of every greater B either a part or parts. If A and B be prime to one another, a A fhall be as many a 4.def.7 parts of the number B, as there are unities in A (as 67.) But if A measures B, it is plain that A is a part of B (as 6 = 18.) b 3,def.7. Laftly,if A and B be otherwise compofed to one another, c the greatest common measure fhall de- 4.def.7. termine how many parts A does contain of B ; as 6 If a number A be a part of a number BC, and another number D the fame part of another number EF then both the numbers together (A+D) fhall be the fame part of both the numbers together (BC+EF,) which one number A is of one number BC. For if BC be refolved into it's parts BG, GC, equall to Ajand EF alfo into its parts EH,HF equall to D; 4 the number of parts in BC shall be equall a hyp. to the number of parts in EF.Therefore fince A+D b=BGEH: GC+ HF,thence A+D fhall bronf. and be as often in BC EF,as A is in BC,which was to be Demonftrated. Or thus. Let ax, and by. then 2 ax, c 2,4x,1, 2 and 2 by.wherefore 2 a+ 2 bx+y. therefore a+b=x+y. w. w. tobe Dem. |