Equations with other Data in them; the which I fhall here omit pursuing, and leave them for the Learner's Practice. Sect. 2. Of Duantities in Geometrical Proportion. GEOMETRICAL Proportion continued has been already defined in Sect. 2. Chap. 6. Part 1. And what is there faid concerning Numbers in may eafily be applied to any Sort of Homogeneal Quantities that are in ÷. The moft natural and fimple Series of Geometrical Proportionals, is when it begins with Unity or 1. As I a a a aaa.aaaa. a'. a, &c. in÷ For I a::a: a a ¦ ¦ a a : a a a ¦ ¦ a à à ; a a a a, &c. That is, when all the middle Terms betwixt the two Extreams are both Confequents and Antecedents, that Series is in Geometrical Proportion continued. Therefore in every Series of Quantities in all the Terms except the laft are Antecedents; and all the Terms except the firft are Confequents. But univerfally putting a the firft Term in the Series, and the Ratio, viz. the common Multiplier, or Divifor; then it will be I. In any of these Series it is evident, that if three Quantities are in, the Rectangle of the two Extrems will be equal to the Square of the Mean; as in thefe, a: a. a, here a xar =aexal, aac. &c. II. If four Quantities are in the Rectangle of the Extreams will be equal to the Rectangle of the Means. As in thefe, a.ae acea eee; here a xa e3a e × g ṛ e. e ee a a a ; here alfo a x -- X a a &c. Confequently, If there are never fo many Terms in the Series of, the Rectangle of the Extreams will be equal to the Rectangle of any two Means that are equally diftant from thofe Ex III. If never fo many Quantities are in it will be, as any one of the Antecedents is to it's Confequents; fo is the Sum of all the Antecedents, to the Sum of all the Confequents. @:ae::a+ae+ace+ae3 + ac2: ac+a¢e + ac3 + ac2 +aes xa+ae+acc+ac3 +ac1. That is, the Rectangle of the Extreams is equal to the Rectangle of the Means; per Second of this Sect. Note, The Ratio of any Series in viding any of the Confequents by it's Antecedent. Thus, a) a (e increafing, is found by di Orae) ace (e, &c.' But if the Series be decreafing, then the Ratio is found by dividing any of the Antecedents by it's Confequent. CONSECTARY. Thefe Things being premifed, fuch Equations may be deduced from them, as will folve all fuch Questions as are usually propofed about Quantities in Geometrical Proportion. In order to that, = the firft Term. the common Ratio. let y the laft Term. } as before. S the Sum of all the Terms. Then Sy And S the Sum of all the Antecedents. Analogy. 1a: ae:; S-y: S-a per III. of this Sea. 24 3 S-a es 3+ey 4 Stey—a=is -ey 4-S S 5ey a=es-s ye 6 S, the Sum of all the Series. 4+a10 S+ey=es+a 10- -eSIIS +ey-eS=a, the firft Term. Note, The. fet in the Margin at the fecond Step, is instead. of ergo; and imports that the Rectangle of the two Extreams in the first Step, is equal to the Rectangle of the Means. And fo for any other Proportion. Sect. 3. Of harmonical Proportion. HARMONICAL or Mufical Proportion is, when of three Quantities (or rather Numbers) the firft hath the fame Ratio to third, as the Difference between the firft and fecond, hath to the Difference between the fecond and third. As in thefe fol lowing. Suppose a, b, c, in Mufical Proportion. Then1a:c::b-a:c-b 42cb=ca=as-ba 2+ If there are four Terms in Musical Proportion, the first hath the fame Ratio to the fourth, as the Difference between the firft and fecond hath to the Difference between the third and fourth. That is, let a, b, c, d, be the four Terms, &c. Then I a: d::ba: d-c 1.2db da-da-ca 2-da 3 db 1= 2 da ca Of Proportion Disjunct, and bow to turn Equations inte Analogies, &c. And PROPORTION Disjunct, or the Rule of Three in Sect. I SECT. I. F four Quantities, (viz. either Lines, Superficies, or Solids) be Proportional: the Rectangle comprehended under the Extreams, is equal to the Rectangle comprehended under the two Means. (16 Euclid 6.) For Inftance, Suppofe, a.b.c.d. to represent the four Homogeneal Quantities in Proportion, viz a:b:c: d; then will ad bc. For fuppofe b 2 a, then will d2c, and it will be a : 2 a::c: 2 c. Here the Ratio is 2. But 4 × 2 c = ax c. viz. 2 ca 2 ac. Or fuppofe 63a, then will d= 3c, and it will be a 3a::c: 3c. Here the Ratio is 3. But a x 3c =3axc. viz. 3 ca=3ac. Or univerfally putting e for the Ratio of the Proportion, viz. making bae, then will dce, and it will be a: ae::c:ce. But a x cea ex c, viz. ace =aec. Confequently, a d=be which was to be proved. : Whence it follows, that if any three of the four Proportional Quantities be given, the fourth may be eafily found; thus, If four Quantities are Proportionals they will alfo be Proportionals in Alternation, Inverfion, Compofition, Divifion, Converfion, and Mixtly. Euclid 5. Def, 12, 13, 14, 15, 16. That |