vide by m, and then multiply by #, (which will make the Work eafier.) For 'tis plain, that the Ratio a:f, divided by 5, gives a : b; and this multiplied by 2, gives a : 6, the of a: f. Or, multiply af by 2, and call the Product a:l; then is a:ff:l; therefore betwixt f: there fall 4 Means in the fame Ratio, as that of ab, making in all this Series, a:b:cid:e:f:g:b:i:k:l. Now here 4:c: egil are; therefore ac is the of a: 1, that is, the of double 5 5 5 of a:f; the fame as the Double of a:b, which is the of af. And this fhews the Advantage of beginning firft with the Divifion. 5 3 For a particular Example. Suppofe 8: 27 multiplied by (or divided by 2)` the Product is 49; as in this Series 8: 12: 18: 27, where 8:12::2:3, and 8:18:: 4:9; and as 2 : 3 is of 8: 27, fo 4: 9 is of it: But had we first 3 3 multiply'd, the Question would have appear'd in this Series, 64: 96: 144 : 216: 324: 486: 729; wherein 64: 729 :: 82: 272, the Double of 8:27; and 64: 144 (or 4:9; for 64: 144: 4:9) is the Part of it; because 64:144 : 324: 729 are 1. To conclude, 3 The Senfe of this Cafe may be refolved into this Problem, viz. To find the Ratio of the Extremes of a Series 1, containing a given Number of Terms, (viz. 1 more than the Numerator of a given Multiplier, or the Denominator of a Divifor) and whofe Ratio is fuch, that another Series may be found in the fame common Ratio, whofe Number of Terms is another Number given, (viz. 1 more than the Denominator of the given Multiplier, or Numerator of the Divifor) and whofe Extremes are in a given Ratio, (viz. that propos'd to be multiplied or divided.) The preceding Series and Explication fhew manifeftly that this is the true and proper Meaning of multiplying or dividing a Ratio by a Fraction. Cafe 3. To divide one Ratio by another, both of one Species, i. e. having two Ratios of one Species given, to find how oft the one is contained in the other; or to find that Number by which the one being multiplied (according to Cafe 1, Multipli cation of Ratios) the Product shall be equal to the other given Ratio; and if the Divifor is not an aliquot Part of the Dividend, we are to find the greatest Number of Times it is contained in it, and alfo the Ratio that remains over. Rule. Subftract the Divifor from the Dividend (by Subftraction of Ratios) and the fame Divifor from the Remainder, and the fame Divifor again from the last Remainder, and fo on continually, till the Remainder be a Ratio of Equality; and then the Number of Subftractions is the Number fought; in which Cafe the Divifor is an aliquot Part of the Dividend: Or, till the Species of the Ratio in the Remainder is different from that of the given Ratios; and then the Number of Subftractions lefs is the greatest Number of Times the Divifor is contained in the Dividend; and the laft Remainder but one, is the Ratio that is contained in the Dividend more than fo many Times the Divifor. Thus, if the Series ab:cd: e is, then is a: b contained in a :e four Times: For ab taken from a: e, leaves be; and from this taking a:b (or b:c) the Remainder is ce; and from this take a b (or : d) the Remainder is de; and from this take ab (or d: e) the Remainder is 1:1; and the Number of Substractions being 4, this is the Quote. Again: Suppofe ab: c:d arel, but d: e a different Ratio; then fuppofing the Series to increase from a to e; 'tis plain that de must be greater than cd; or e must be less than a third Proportional to c:d; for if it were greater, we should have another Term betwixt d and e in the preceding Ratio. Now then, a: b from a: e, leaves be of the fame Species. Again, ab (or b: c) from be leaves c:e of the fame Species; ab (or c: d) from ce, leaves die of the fame Species. Laftly, ab from de makes the Remainder db: ae, which is of a different Species; for a : b and d:e, being both Ratios of the leffer to the greater; and a proper Fracti on leffer than 4, the Quote db e is an improper Fraction. If the Series decrease from a to e, then is e greater than a true third Proportional to cd; fo that when a : 6 is taken from d : e, the Remainder Cafe greater than d db de a is a proper Fraction, because is in this b Wherefore the Number of Substractions, lefs 1, viz. 3, is the Quote; and the Remainder of the Divifion is the Ratio de. BOOK V. Containing these following SUBJECTS, VIZ. 1. Of the Compofition and Re-IV.Of Infinite Decimal Fractions.. Jolution of Numbers. II. Of Figurate Numbers. HI. Of Infinite Series. V. Of Logarithms. VI. Of CHAP. I. Of the Compofition and Refolution of Numbers: Or, The Doctrine of Prime and Compofite Numbers; with that of the Commenfu rability and Incommenfurability of Numbers. I. Ο §. 1. Containing the General Principles and Theory. DEFINITION S. NE Number is faid to measure, or be a Measure of another, when it is contained in it a certain Number of Times precifely; fo that being taken out of it as oft as poffible, there fhall nothing remain over. Thus, 4 meafures 12 ; becaufe it is contained in it precifely 3 Times. Obferve alfo, that one Number is faid to measure another by that Number which is the Quote: So 4 meafures 12 by 3; and reciprocally, 3 meafures 12 by 4: And hence any Number with the Quote, by which it measures another, may be called the reciprocal Measures of that Number. COROLLARIES. 1. Every aliquot Part of a Number measures it; and every Number which meafures another, is an aliquot Part of it. 2. Unity meafures every Number by that Number it felf; and every Number meafures it felf by Unity; and these are the greatest and least Measures of any which are also reciprocal Measures. any Numbers; II. II. A Number is called the Common Measure of two or more Numbers, when it meafures each of them: So 3 is a Common Measure of 6, 9, 12. And if it's the greateft Number that measures them, it is called their Greatest Common Meafure; as Unity is their leaft. III. A Number is called a Prime Number, which has no Measure, but it felf and Unity; as 2, 3, 5, 7; and which confequently is the Product of no other Numbers. IV. A Number is called a Compofite Number, which has fome Measure befides it felf and Unity; and which confequently is the Product of fome two other Numbers: For every Measure has its Reciprocal, and their Product is the Number measured by them, from the Nature of Divifion. Thus, 3 meafures 12 by 4, and 3 Times 4 is 12. V. Two or more Numbers are faid to be Commenfurable, when they have fome common Measure befides 1. Thus, 6: 9 are Commenfurable, because 3 measures them both; and 5, 10, 15, because 5 measures them all. VI. Two or more Numbers are faid to be Incommenfurable, when they have no common Measure befides 1, as 3, 4; or 4, 5, 6. Such Numbers are alfo faid to be Prime to one another, or among themfelves; though none of them be really Prime in it felf. COROL. Two or more Prime Numbers are Incommensurable, because they have no Common Measure but 1. And hence again; if feveral Numbers, A, B, C, &c. are Commenfurable, no two of them can be Prime Numbers; and if one of them is a Prime, it must be the common Measure of the Whole, clfe they have no Measure,, fince that Prime has no other Measure befides 1. SCHOL. Tho' Unity is a common Measure of all Numbers, yet the Notion of Compofition and Commenfurability is limited fo as to exclude 1 from being a Measure: For fince I measures all Numbers, if this were admitted, there would be no fuch Diftinctions as Prime and Compofite, Commenfurable and Incommenfurable. If we take Compofition in the largest Senfe, then. Unity is the only Number which we can call Simple; all others being Collections or Compofitions of Units: But this Confideration, is too general and fimple to be of any Ufe in difcovering the Properties of Numbers; and therefore the Compofition here treated of, is that particular Kind which depends. upon Multiplication, taken in its more proper and ftrict Senfe, as applied to the Repetition of Numbers, or multiplying them by a Number greater than Unity, because Unity apply'd as a Multiplier, makes no Alteration of the Number to which it is. apply'd. Obferve alfo, That Integral Numbers only make the Subject of this Chapter : For if Fractions were admitted, then there is no Number, either Integral or Fractional, but fome Fraction will meafure it. For Example; Let A be any whole Number; take any Fraction whofe Numerator is 1, as, it will meafure A; the Quote A L n being #A: Again, let be any Fraction, 'tis meafurable by this Fraction АВИ, Quote being n. Again obferve, that the Distinction of Fractions into Simple and Compound, explained in Book H. is nothing like this Distinction of Prime and Compofite; even though Multiplication is concerned in that Compofition; for that is merely a Diftinction of two different Forms of expreffing the fame Quantity: Thus, 2 of 4, and 8 are but the fame Thing differently conceived and expref3 ་ 5 15 fed: So that if we take the Notion of Compofition in General, as the Effect of Mul tiplication, tiplication, then 8 15 is a compound fractional Number, tho' expreffed in a fimple Form: And in this Senfe we can call that only a Simple Fraction which is not the Product of two real Fractions, proper or improper (excluding fuch improper ones whofe Value is an Integer, and not a mix'd Number) and fuch only are all Fractions, whofe Denominators are Compolite Numbers. Example: is a Simple Fraction, both in Form and in its Nature; because no two Fractions produce it, or it is not the Fraction of a Fraction; but this is Simple in its Form, and in its Nature Compound; for it is 1 of 1, and 15 8 2 = of 4 or 5 5 3 4 of: And this Simplicity and Compofition is what belongs 15 5 ༢) 5 3 to Fractions, but has nothing to do in the Subject of this Chapter. VII. A Number is called Even, which is Measurable by, or is a Multiple of 2; as 2, 4, 6, &c. VIII. A Number is called Odd, which is not Measurable by, or is not a Multiple of 2; as 3, 5, 7, &c. COROLLARIES. 7. An odd Number divided by 2, leaves 1 of Remainder. 2. Take the natural Progreffion.2.3.4, &c. and beginning at 2, take every other Number, i. e. take one and leave the next continually, and fo you have the whole Series of even Numbers, 2.4.6.8.10, &c. for the Series, 1, 2, 3, &c. having for the common Difference, the Difference of any Term, and the next but one, is 2; confequently beginning at 2, and taking every other Term, we have a Series differing by 2 which beginning with 2, is therefore the Series of Multiples of 2, i. e. of all even Numbers. Hence again, if we begin at 3, and take every other Term, as 3, 5, 7, 9, 5c. we have the Series of odd Numbers; which proceeds alfo by the common Difference of 2. 3. If we take the natural Progreffion, 1. 2. 3, &c. and double each Term of it, the Series of Products is the Series of even Numbers; because it is the Series of Multiples of 2. And taking the fame natural Progreffion, if we take the Sums of every two adjacent Terms, thus, 1+2: 2+3: 3+4, &c. thefe make the Series of odd Numbers 3 57, 5c. for the firft, 1+23 is the first odd Number, and the Series proceeds by the conftant Difference of 2; becaufe every two adjacent Sums have one Part common, and the other Parts are either two adjacent odd Numbers, or two adjacent even Numbers; which differing by 2, therefore the Series of Sums differ by 2; and because the first is 3, they muft make the Series of odd Numbers. 4. 1 added to any even Number or fubftracted from it, makes the Sum or Remain der the next greater or leffer odd Number; and I added to or fubftracted from any odd Number, makes the Sum or Difference the next greater or leffer even Number. Again, 2 added to or fubftracted from any even or odd Number, gives the next greater or leffer Number which is also even or odd. 5. All even Numbers have 2, 4, 6, 8, or o, in the Place of Units if they exceed 8; for they proceed from the continual Addition of 2 to it felf, and to every fucceeding Sum; but the firft of them are thefe 2.4.6.8. 10, and confequently the fame Figures muft circulate continually in the Place of Units. Again, all odd Numbers above 9, have in the Place of Units, one of thefe Numbers, 3, 5, 7, 9, or 1; for all odd Numbers proceed from the conftant Addition of 2, firft to 1, and then to the Sum, making the first 5 odd Numbers thefe, 3, 5, 7, 9, 11, whence it's plain, that the fame Figures will continually circulate in the Place of Units. |