Ch. XIV. I. Logarithms of Accordingly, the Half of 167 is 831, to which adding, the Sum is 84, the Number of Sheep taken by the first Company; and then the first Remainder was 83. Again, the Half of 83 is 41, to which adding, the Sum is 42, the Number of Sheep taken by the second Company; and fo the fecond Remainder was 41 Sheep. Laftly, the Half of 41 is 20, to which adding, the Sum is 21, the Number of Sheep taken by the third Company; and fo the third Remainder is 20 Sheep according to the State of the Question. CHA P. XIV. Of the Use of Logarithms. I Multiplication and Division of larger T has been (†) above observ'd, that the great Use Numbers, as alfo the Extraction of the in Multi Root of any Power, is render'd much plication, Divifion, more easy by the Help of Logarithms, and the I have therefore referv'd this laft ChapExtracti- ter to explain therein the Ufe of Loga rithms. on of Roots. 2. Loga rithms, what. Logarithms are artificial Numbers, proceeding in Arithmetical Proportion, as the natural Numbers, to which they are applied, (+) Chap. 5. §. II. and Chap. 6. §. 8. and Chap. 12. §. 15. applied, do proceed in Geometrical Pro- Ch. XIV. portion, viz. garithm As every Number confifts of two Parts, 3. a Numerative and Denominative; fo Every Loevery Logarithm confifts of two Mem-confifts of bers, diftinguish'd one from the other twoParts. by fome feparating Mark, whether full Point or Comma, &c. That Member of the Logarithm, which ftands on the Right-hand of the feparating Mark, has respect to the numerative Part of the Number, to which the Logarithm belongs; the other Member on the Lefthand, has respect to the denominative Part of the faid Number. Namely, hereby the Denomination or Place of the last (i. e. left-hand) Figure, and confequently of all the other Figures in the Number, is indicated or fhewn; whence this Member of the Logarithm is peculiarly ftil'd the (*) Index. For Inftance: The (*) It is otherwife call'd the Characteristick. Index Ch. XIV. Index [o] being affix'd to a Logarithm, denotes, that the laft Figure of the Number, to which the Logarithm anfwers, is nothing diftant from (i. e. is in) the Place of Units, and confequently, that the faid Number is one of the Digits. The Index (1) denotes the laft Figure of its correfpondent Number, to be diftant one Place from the Place of Units, i. e. to be in the Place of Tens; and confequently, the Number it felf to be either Ten, or fome Number between 10 and ico. And fo of other Indices. 4. Loga rithms, Hence all Numbers, which have the How the fame Denominative, but not the fame Agreement of numerative Part, (fuch as are all Numbers from 1 to 10, or from 10 to 100, c.) will likewife have Logarithms, answers to the Agree- whofe Indices will be the fame, but not ment of their other Members. And on the other Side, all Numbers, which have the fame numerative Part, but not Denominative, will also have the fame Logarithm, excepting only the Inftance: Numbers. Numb. of fame De- but not of fame Nu merative. 256 257 258 different Index. For Their respective 5. Of the Logarithms mals. If a Number be purely or wholly a Decimal, then to the Logarithm thereof is to be affixt a negative Index, fhewing of Decithe Distance of its firft (i. e. left-hand) fignificative Figure from the Place of Units. Thus, the Logarithm of the Decimal 1256, is T. 40824; the Logarithm of the Decimal 10256, is 2.40824, &c. Logarithms are of great Ufe in multi- 6. plying or dividing larger Numbers; for- Multipliafmuch as the Logarithms of the two Divifion, Factors in Multiplication, being added to- how fhorgether, make up the Logarithm of the LogaProduct; and in Divifion, the Logarithm rithms. of the Divifor, being fubftracted from the Logarithm of the Dividend, leaves the Logarithm of the Quotient. For Inftance : cation awd ten'd by Ch. XIV. 7. The making of In like manner, the Logarithm of any Root, being multiplied into the Index of any Power, gives the Logarithm of the faid Power: That is, the Logarithm any Power, of a Root, being doubled, gives the Extracti Logarithm of the Square; being tripled, on of any gives the Logarithm of the Cube. And fhorten'd fo on. Thus, because the Logarithm of by Loga- 9, is 0.95424; therefore, o . 95424 × 2= and the Root; how rithms. 1.90848, the Logarithm of 99 or 81. And 0.95424×3=2. 86272, the Logarithm of 9c or 729. Namely, And therefore, on the other Hand, the Logarithm of any Power, being divided by the Index of the faid Power, will give the Logarithm of the Root. For Inftance: The Logarithm of the Square 81, 90848. Wherefore, 2) 1.96848 (0.95424, the Logarithm of 9. In like manner, the Logarithm of the Cube 729, is 2.86272. Wherefore, 3) 2.86272 (o. 95424. Only here it is to be noted, that although in common Divifion of Integers, when the Divifor is bigger than the left-hand Figure of the Dividend, the |