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Chambers, Ephraim, 1680 (ca.)-1740 / Cyclopædia, or, An universal dictionary of arts and sciences : containing the definitions of the terms, and accounts of the things signify'd thereby, in the several arts, both liberal and mechanical, and the several sciences, human and divine : the figures, kinds, properties, productions, preparations, and uses, of things natural and artificial : the rise, progress, and state of things ecclesiastical, civil, military, and commercial : with the several systems, sects, opinions, &c : among philosophers, divines, mathematicians, physicians, antiquaries, criticks, &c : the whole intended as a course of antient and modern learning

(1728)

Locustae - Lysiarcha, pp. 466-477 PDF (10.9 MB)

Page 467

( 467 ) or the Direaion of the Veffel with regard to the Points of the Compafs. In the third, the Number of Knots run off the Reel each time of heaving the Log. In the fourth, the Wind that blows: and in the fifth, Obferva- tions made of the Variation of the Compafs, &tc. LOGARITHMIC, or LOGISTIC CURVE, a Curve generated by the equable Motion of the Radius of a Circle, thro' equal Arcs of the Circumference ; while at the fame time a Point in that Radius is fuppofed to move from the Arc towardG the Centre, with a Retardation of Motion in a Geometrick Proportion. As fuppofe there be a Quadrant of a Circle, B C A, (Plate Analjis, Fig.t r.) and any equal Divifions in the Arc, as A F- Ff-ff; U!c- with five corresponding Radii, as fuppofe C A, C F, Cf, &c. whofe Parts or Portions C I, C a, C a, &c. are geometrically Pro- portional; then if a Line, as I, a, a, 6, d, C, be drawn thro' thofe Points, it will be the Logarithmic or Logi~fic Spiral. LOGARITHMS (from Aot(£ ratio, and det9u- nu- merus) are ufually defined Numerorum Propertionalium equi- dtfferentes Comites; but this Definition Dr. Halley and Stife- lius think deficient, and more accurately define them, The Indices or Exponents of the Ratio's of Numbers ; Ratio being confider'd as a Quantity fui generis, beginning from the Ratio of Equality, or I to i - o; and being affirma- tive when the Ratio is increafing, and negative when it is decreafing. The Nature and Genius of Logarithms will be eafily conceiv'd from what follows. A Series of Quantities increafing or decreafing accor- ding to the fame Ratio, is call'd a Geometrical Progref- lion ; e.g. I. 2. 4. 8. 16. 32- &c. A Series of Quantities increafing or decreafing according to the fame Difference, is called an Arithmetical Progreftion ; e. g. 3. 6. 9. I 2. I 5. IS. 24. Now if underneath the Numbers proceeding in a Geometrical Ratio, be added as many of thofe pro- ceeding in the Arithmetical one ; thefe laft are call'd the Logarithms of the firfi. Suppofe v. g. two Progrefflons: GeomeI. 2. 4. 4. 8. i6. 32. 64. Iz8. 256. 512 Arithmet.o. I. 2. 3. 4. 5. 6. 7. 8. 9 Logarithms. o will be the Logarithm of the firfl Term, viz. I i 5 of the 6th, 32t 7 the Logarithm of the 8th, i28, &c. Theor. 1. if the Logarithm of Unity be o, the Logarithm of the FaFum or ProduHl will be equal to the Sum of the Logarithms of the Fa.ors. pem. For as Unity is to one of the Faaors, fo is the other Fadtor to the Produc&. So that the Logarithm of the Product is a fourth equidifferent Term to the Loga- ritbm of Unity and thofe of the Fadors: but the Loga- rithm of Unity being o, the Sum of the Logarithms of the Faacors muft be the Logarithm of the Faaum or Pro- du&. q. e. d. Corol. i. Since the Faaors of a Square are equal to each other, i. e. a Square is the Fadum or Product of its Root multiplied into itfelf; the Logarithm of the Square will be double the Logarithm of the Root. Corol, - In the fame manner it appears that the Loga- rithm of the Cube is triple, of the Biquadrate, quadruple; of the fifth Power, quintuple; of the fixth, fextuple, tc. of the Logarithm of the Root. Carol. 3. Unity, therefore, is to the Exponent of the Power, as the Logarithm of the Root to the Logarithm of the Power. Corol. 4. So that the Logarithm of the Power is had, if the Logarithm of the Root be multiplied by its Expo- nent; and the Logaritbm of the Root is had, if the Loga- ritbm of the Power be divided by its Exponent. Schol. Hence we derive one of the great Lfes of Loga- rithms, which is to expedite and facilitate the Bufinefs of Multiplication and ExtradJion of Roots ; the former of wbicb is here perform'd by mere Addition, and the latter by Multiplication. Thus 3, the Sum of the Lomarithms i and 2, is the Logarithm of 8, the Produ& of 2 and 4. In like manner 7, the Sum of the Logarithms 2 and 5, is the Logarithm of i28, the Produdt of 4 and 32. Again, L OG 3, the Logarithm of the Square Root 8, is half the Lng*- rithm of 6, the Square Root of 64 i and z, the Logarithm of the Cube Root 4, is fubtriple the Logarithm 6 of the Cube 64. Theor. II. If the Logarithm of Unity be a, the Lozarithm of the Quotient will be equal to the Dijference of the Lo- garithms of the Divifor and Dividend. Dem. For as the Divifor is to the Dividend, fo is Unity to the Quotient; therefore the Logarithm of the Q(uotient is a fourth equidifferent Number to the Logarithms of the Divifor, the Dividend, and the Logarithm of Unity. The Logarithm of Unity therefore being o, the DifFerence of the Logarithm of the Divifor and that of the Dividend, is the Logarithm of the Quotient. q. e. d. Schol. Hence appears another great Advantage of Lo. garithms, viz. their expediting the bufinefs of Div zon, and performing it by a bare SubftraFlion. E.g. 2 the Dif- ference between 7 and 5, is the Logarithm of the Quo- tient 4 out of 128 by 32. In like manner, 5 the Diffe- rence between 8 and 3, is the Logarithm of the Quotient 32, out of 256 by 8. An Example or two will render the Ufe of Logarithms in Multiplication, Divifion, Zfc. obvious. Num. Log. Multiply 68 1.83250 by I2+ 1.07918 8X6 2.91i68 9 0.95424 9 0.9 5424 Num. Log. Divide 816 2.91168 9 by I2 1.07918 9 68 1.83250 Sq-.8 9 0.954'4 9 0.95424 -99 0595424 Sq.8 1 2)1 .9g848(0-95424 Oq.R. Cube 729 3)2.8627z(o0934Z4 Cube R. The Properties of the Logarithms hitherto mention'd, and their various Ufes, are taken notice of by Stbfebus: but come all far Ihort of the Ufe of Logarithms in Trigono- metry, firfl difcover'd by the Lord Neper. To find the Logarithm of any Number, and to conjlruFe a Canon of Logaritbms for Natural Numbers. i. Becaufe I. ro. zoo. ioco. 10000. sc. conflitute a Geometrical Progrefflon, their Logarithms may be taken at pleafure: To be able, then, to exprefs the Logarithms of the intermediate Numbers by Decimal Fraclions, take o.oooosooo, i.00000000, 2.oooooooo0 , 3.C000000, 4.00000000, TOc. 2. 'Tis manifefI that for thofe Numbers which are not contained in the Scale of Geometrical Progreffion, the jufl Logarithms cannot be had: yet may they be had fo near the Truth, that as to Matters of Ufe they Ihall be altogether as good as if flridly juft. To make this ap- pear, Suppofe the Logarithm of the Number 9 were re- quired : between i.oooaooo and i0.0000000 find a Mean Proportional, and between their Logarithms o.oocooooo and i.ruijocco an Equidifferent Mean, which will be the Logarithm thereof, that is, of a Number exceeding Threeby I 00007-7a7, and therefore far remote from Nine. Between 3 and lo therefore find another Mean Propor- tional, which may come fomewhat nearer Nine; and between io and this Mean, another fill; and fo on be- tween the Numbers next above and next underneath Nine, till at lafi you arrive at 9-00000000, that is, 9 f00fOOoo which not being one Millionth Part from N ine, its Logarithm may, without any fenfible Error, be taken for that of Nine itfelf. Seeking then in each Cafe for the Logarithms of the Mean Proportionals, and you will at lafi have 0.9542 51, which is exceedingly near the true Logarithm of Nine. 3. If in like manner you find Mean Proportionals be- tween I.ococo0o and 3.1622777, and afiign convenient Logarithms to each, you will at length have the Logarithm of the Number 2, and fo of the rea. Mean L O G

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