COROLLARY XVIII.

287. The least Integral Monomes of their Number of Places has been fhewn to be 10=9+1, 100 = 99 + 1, 1000 = 999 + 1, 10000 = 9999 + 1 &c. (In. 271) whofe Powers are as follows (In. 280.)

Whence it appears by Infpection, that counting from Unit's Place, there can be but one Place in the Root for every Period of two Places in the Square, of three Places in the Cube, of four in the Biquadrate, &c. and one Place for what remains over and above; which in each Power is alfo to be taken as a Period of it felf.

HYPOTHESIS IV.

288. The Periods of Powers are diftinguished by drawing Lines over the Head of each; Ex, gr. the Number 68719476736 if confidered as a Square confifts of fix Periods 6 87 19 47 07 36, if as a Cube of four Periods 68 719 476 736, if as a Biquadrate of three 687 1947 075, if as a ffrft Surfolid of three 6 87194 76736, if as a fixth Power of two 68719 476736, &c.

COROLLARY XIX.

289. For every Cypher, or Fractional Place poffeffing the Right-hand in any Number confidered as a Root, there must be two in its Square, three in its Cube, four in its Biquadrate, &c.

COROLLARY XX.

290. The Root of the next Homologous Power to the highest Period in every given Refolvend is the first Figure of the Root required, and of the fame Place from Unity that the highest Period is from the Period of Units. Ex. gr. in the Square Refolvend 6 87 19 47 67 36 the Periods are fix, viz. 60000000000 + 8700000000+ 19000000 + 470000 + 6700+ 36, confequently the Root confifts of fix Places (In 284) i. e. if the loweft Place in the lowest Period be Units Place, then the highest Place in the Root is the Place of Hundred Thousands (In. 254) and becaufe by the Table (In. 349) the next greatest Square under 60000000coo is 40000000000 whofe Root is 200000, therefore 200000 is the firft Figure in the Root. Again in the Cube Refolvend 68 719 476 736 the Periods are four, viz. 6800000000 + 719000000 +476000 + 736 confequently the Root confifts of four Places (In. 284) i. e.

if the lowest Place in the lowest Period be Units Place, then the highest Place in the Root is the Place of Thousands (In. 254) and because by the Table (In. 349) the next greatest Cube under 68000000000 is 64000000000 whofe Root is 4000, therefore 4000 is the firft Figure in the Root. In like manner in the Biquadrate Refolvend 687 1947 6736 the Periods are three, viz.. 68700000000+ 194,0000 + 6736, confequently the Root confifts of three. Places (In.. 284) i. e. if the loweft Place in the lowest Period be Units Place, then the higheft Place in the Root is the Place of Hundreds (In. 254) and becaufe by the Table (In. 349) the next greatest Biquadrate under 68700000000 is 62500000000 whofe Root is 500, therefore 500 is the first Figure in the Root.

CHA P. III.

Of Mixed Polynomes, and the Tables of Weight, Coin, and MEASURES.

DEFINITION X.

291.

191.1

Call that a mixed Polynome which expreffes different Denominations of Quantity according to the Table of Coin, Weights, Measures, &c. to which it belongs.

DEFINITION XI.

292. The Standard of the currant Coin of England is called Sterling or Efterling, by which is meant the Weight of an English Silver Penny, which ought to be equivalent in Weight with 32 Grains of well dried Wheat taken out of the middle of the Ear: The fourth Part of this is termed a Farthing marked gr. for Quadrans, which is the leaft Denomination of English Coin; as in the following

TABLE..

4 grid Denarius, a Penny 48 gr=12d = is Solidus, a Shilling

oss. ood (1 Crown! An Angel..

10s. Ood

06s. 80dis A Nible.

960 qr=240d. = 20s = 11 Libra, a Pound Sterling 1,535 04d A Mark.

D

17.015,ood) (A Guinea.

There

293. Troy Weight is that whereby Corn, Bread, Liquors, Gold, Silver, Jewels, Amber, Ele&tuaries, &c. are weighed the least Denomination of this Weight is an Artificial Grain, Gr. 24 of which are equal in Weight to 32 Grains of Wheat.

294. Apothecaries Weight is a different Table of Troy Weight, by which Apothecaries compound their Medicines.

4

ᄒ

0.3.0.12

0.2.1.04

&c. &c.

3 Э gr. 3

&c.

16

12 =

&c.

295. Averdupois Weight is that whereby Flesh, Butter, Cheefe, Tallow, Salt, Flax, Hemp, Wax, Pitch, Tar, Rofin, Copper, Tin, Steel, Iron, Lead, Tobacco, &c. and in general all Kinds of Grocery Wares, and whatsoever is fubject to waste, are weighed. The Pound Averdupois Mr. John Ward of Chester, fays, he found by a very nice Experiment to be equal to 14 oz. 11 p.w. 15 Grains Troy. The leaft Denomination of this Weight is a Dram, Dr.

296. The Denominations of Wool-Weight are as follows, the leaft Denomination being 1 lb. Averdupois.

The TABLE.

2:

28. lb.

182 lb

182 lb. = 26 Cl. =

364 lb. = 52 Cl. =

4368 lb. = 624 Cl.

13 St. = 6 Tdd. = 1 W. Wey.

26 St. = 13 Tdd. = 2 W. = 1 S. Sack.

= 312 St. = 156 Tdd. =24W.=12S.=1 Laft or Wey.

DEFINITION XVI.

297. The leaft Denomination of Long Meafure is an Inch In. which is fuppofed to be equal in Length to three Corns of Barly well dried, and taken out of the middle of the Ear.

45 20 60 26 72 198 88

32

2 I Span.

3=1}= 1 Foot.

I Cubit.

9 4 3 2 1 Yard.
15=114= 5=34=2;= = 1 Ell.
20 15 6;= 5= 3;===
24 18:

6=

3}=}}=}}= 1 Pace. 6 4 2=1}= 1;= 1 Fathom. 66= 49;= 22=16;= 11= 5;= 4}=3,!= 24=1 Perch. 7920=3520=2640=1980=880-660=440=220=176=132=110=40=1 Furi..

8 Furlongs
3 Miles

1 Mile.

Si League,

s Degree,

} make {

Part of the Earth's Circumference.

DEFINITION XVII.