HYPOTHESIS XVIII.

205. The Sign of Arithmetical Progreffion is

HYPOTHESIS XIX.

206. If the Species / be put indefinitely for the leaft Term of any Series in ,g for the greateft, d for the common Difference, n for the Number of Terms, and s for the Sum of the whole Series; then every encreasing Series will be thus reprefented I, 1+d, l + 2d, l + 3d, 1 + 4d, l+ 5d, &c. according to n the Number of Terms: And every decreafing Series, thus, g, g-d, 8-2d, g — 3d, 8—4d, g-5d, &c. till we come to the leaft

Term.

COROLLARY XLVIII.

207. The greatest Term of any Series in is equal to the least Term added to as many times the common Difference, as there are Terms in the Series lefs one, i. e. g = 1 + d Xn-1 or g = 1 + nd — d.

COROLLARY XLIX.

208. If 70, then the encreafing Series in will be thus, o, d, 2d, 3d, 4d, 5d, &c. till we come to gnd-d: The decreafing Series thus, nd d, nd 2d, nd-3d, nd- 4d, nd-5d, &c. till we come to lnd —nd = o. Indndo.

COROLLARY L.

209. In any Series of Terms in the Difference between all fuch Terms. as are equally diftant from one another is equal, as confifting of the Common Difference the fame Number of times repeated; Ex. gr. the Difference between the first and third Terms is the fame with the Difference between the fecond and fourth, and between the feventh and ninth, &c. fo alfo the Difference between the second and fixth Terms, is the fame with the Difference between the fourth and eighth, and between the fifth and Ninth, &c. Therefore

CORROLLARY LI.

210. If any Number of Terms be in, the leaft Term the greatest Term, and any two middle Terms which are equally distant from the least and greateft, will be four Terms in Arithmetical Proportion (In. 167.) and confequently the Sum of the two Extreams will equal the Sum of any two equidiftant Means. (In. 174.)

COROLLARY LII.

then the leaft Term, the

211. If any odd Number of Terms be in greatest Term, and the middle Term will be three Terms in Arithme

tical Proportion (In. 167), and confequently the Sum of the two Extreams will be double the Mean (In. 176.)

THEOREM VII.

212. In any Series of Terms in, if the Sum of the greatest and least Terms be multiplied into One half the Number of Terms, the Product will be twice the Sum of the whole Series; i. c. x+8=s.

72

2

Demonftration.

1+g is equal to the Sum of every two equidiftant Means (In 209.) and to Double the middle Term of all, if the Number of Terms be odd. (In 210.) But if every two Terms in the whole Rank be thus fumm'd up, there

S

can be but half so many Sums, or +g, as the Number of Terms n, :. the Sum of all the Terms in the Series must equal times g: i. e. s ===

n

2

12

2

213. Therefore the Sum of every Rank of Laterals beginning with Unity (ie. when l = d= 1, confequently gn) will be 2

COROLLARY LIV.

nn + n

215. Geometrical Progreffion is when a Series or Rank of Homogeneous Terms do encreafe or decrease by the fame Geometrical Ratio, as,

I, 2, 4, 8, 16, 32, 64, 128, 256, &c.

256, 128, 64, 32, 16, 8, 4, 2, }whofe Nominator is 2.

I.

I, 3, 9, 27, 81, 243, 729, 2187, 6561, &c.

6561, 2187, 729, 243, 81, 27,

9, 3, I

whofe Nominator is 3.

And fo for any other Rank where the Nominator is 4, 5, 6, 7, &c.

SCHOLIUM VI.

216. Obferve that the Nominator of the Ratio here always belongs to the Ratio of leffer Inequality.

HYPOTHESIS XX.

L

217. The Sign of Geometrical Progreffion is

HYPOTHESIS XXI.

HYPOTHESIS XXI.

218. If the Species / be put indefinitely for the leaft Term of any Series in, g for the greateft Term, r for the Nominator of the Ratio, n for the Number of Terms, s for the Sum of the whole Series: Then every encreafing Series will be thus reprefented, l, lr, Ir2, lr3, lrt, lrs, &c. And every decreasing Series thus, &,, 8 8 8 &c. according to n the Num

ber of Terms. Whence

COROLLARY LV.

219. The greatest Term of any Series in is equal to the least Term multiplied into that Power of the Nominator, whofe Exponent is the Number of Terms lefs one; i. e. g = lrn

COROLLARY LVI.

And if g = 1

220. If I then will the encreafing Series in be thus represented, I, r, r2, r3, ra, r3, r6, &c. till we come to gr the decreasing Series will be 1, 7, 1, 7, 74, 7, &c. or which is the fame 3, r-4, r-s, &c. (In. 154.)

thing r, r-',

COROLLARY LVII.

221. Therefore every Scale of Powers is a Rank of Terms in Geometrical Progreffion, whofe firft Term is Unity, and Nominator is the Root. And the Exponents of the Scale are a Rank of Terms in Arithmetical Progreffion whose first Term is o, and Common Difference is Unity.

COROLLARY LVIII.

222. Ín every Series of Terms in, the Ratio between all fuch Terms as are equally distant from one another is the fame.

COROLLARY LIX,

223. In any Number of Terms in, the two Extreams and any two Means which are equally diftant from thofe Extreams are four Terms in Geometrical Proportion (In. 186.) confequently the Product of the two Extreams is equal to the Product of any two equidiftant Means (In. 189.)

COROLLARY LX.

224. If any odd Number of Terms be in the laft Term, the greatest Term, and the middle Term are three Terms in Geometrical Proportion

(In. 186.)

(In. 186.) and confequently the Product of the two Extreams is equal to the Square of the Mean.

THEOREM VIII.

225. In any Series of Terms in, the Sum of all the Terms, except the greateft, multiplied into the Nominator of the Ratio will be equal to the Sum of all the Terms except the leaft; i. e. s=gxr = s = l, or sr-gr=s-1.

Demonftration.

Suppose any Series of Terms in whofe leaft Term is 1, Nominator r, and laft Term g; Ex. gr. lrs, i. e. lrg (In. 218.) Then sg= 1 + lr + lr2 + Ir3 + Ira, and s − 1 = lr + lr2 + lr3 + /r+ + lr3 = rxl+lr+lr2+lr3 + Ir1 = rxs—g (In. 71). But this will always be, let the Terms be many or few (In. 214.): • s −1 = rxs—g (In. 21.) Q. E. D.

226.

CHAP. X.

Of Polygonal Numbers.

DEFINITION LXXIII.

226. POLYGONAL Numbers are fuch as arife from the Addition of a

Series of Numbers in beginning with Unity; and are ftiled,, First, Triangulars, or Trigons, when the common Difference is 1..

Arith. Progr. 1, 2, 3, 4, 5, 6, 7, 8, 9, &c.

Trigons 1, 3, 6, 10, 15, 21, 28, 36, 45, &c..

Secondly, Quadrangulars or Tetragons, when the common Difference is 2. Arith. Progr. 1, 3, 5, 7, 9, 11, 13, 15, 17, &c.

{

Telragons.1, 4, 9, 16, 25, 36, 49, 64, 81, &c.. Thirdly, Pentagons, when the Common Difference is 3. Arith. Progr. 1, 4, 7, 10, 13, 16, 19, 22, 25, &c.. Pentagons. 1, 5, 12, 22, 35, 51, 70, 92, 117, &c. Fourthly, Hexagons, when the common Difference is 4. Arith. Progr.SI, 5, 9, 13, 17, 21, 25, 29, 33, &c. Hexagons. 1, 6, 15, 28, 45, 66, 91, 120, 153, &c.. &c. &c.

DEFINITION LXXIV.

&c.

227. The Side or Root of a Polygon is the Number of Terms in the *, which are fum'd up for forming it.

DEFINITION LXXV.

DEFINITION LXXV.

228. The Denominator of the Polygon, which otherwife is called the Number of its Angles, is the firft Polygonal Number of its Kind, next after Unity, from whence each Kind receives its Name; as the Number 3 in a Trigon, 4 in a Tetragon, 5 in a Pentagon, 6 in a Hexagon, 7 in a Heptagon, 8 in an Octagon, &c.

HYPOTHESIS XXII.

229. In the Arithmetick of Polygons put n for the Side or Root of the Polygon, or the Number of Terms in from whence it is formed, d for the Common Difference of the Arithmetical Series, g or G for the greatest Term in the faid, S or P for the Polygon it felf, and D for its Denominator, or the Number of its Angles.

COROLLARY LXI.

230. The Common Difference of the is always equal to the Denominator of the Polygon formed from it, lefs 2; i. e. d D-2.

COROLLARY LXII.

=

231. It appears alfo that every Scale of Polygons (i. e. the Trigon, Tetragon, Pentagon, &c.) formed from the fame Root n is a Rank of Terms in÷ whofe leaft Term is n, and whofe Common Difference is that Trigon which has for its Root n-1; Ex. gr. let n 4, then because the Trigon is 6 whose Root is n13; therefore the Trigon, whofe Root is n = 4, is n+6= 10; the Tetragon is 10+ 6 = 16; the Pentagon is 16+ 6 = 22; the Hexagon is 22+6=28; the Heptagon is 28 + 6 = 34, &c. And the fame for any other Root or Side.

DEFINITION LXXVI.

232. Pyramidal Numbers are fuch as are formed by the Addition of a Series of Polygons after the fame manner as that Series of Polygons were formed by the Addition of a Series of Terms in. And the Sums of thofe first Pyramidals the fame way collected are called Second Pyramidals; the Sums of thofe Second Pyramidals, third Pyramidals, &c. ad infinitum; which in particular are stiled Triangular, Quadrangular, Pentagonal, &c. according as they are formed from a Series of Polygons that are Trigons, Tetragons, Pentagons, &c. Ex. gr. The Genefis of Pyramidals Triangular, are as follows:

Units