Abbildungen der Seite
PDF
EPUB

contrary Way to that by which the Terms, whose Indices are positive, do. These Things premised,

If on the Line AN, both Ways indefinitely extended, be taken A C, CE, EG, GI, IL, on the Right Hand; and alfo AT, III, &c. on the Left; all equal to one another; and if, at the Points I, г, A, C, E, G, I, L, be, erected to the Right Line AN, the Perpendiculars ΠΣ, ΓΔ, ΑΒ, CD, EF, GH, IK, LM, which let be continually proportional, and represent Numbers, whereof A B is Unity: The Lines AC, AE, AG, AI, AL, -АГ, -ΑΠ, respectively express the Distances of the Numbers from Unity, or the Place and Order that every Number obtains in the Series of Geometrical Proportionals, according as it is distant from Unity. So fince AG is triple of the Right Line A C, the Number G H shall be in the third Place from Unity, if CD be in the first: So likewise shall LM be in the fifth Place, fince AL=5AC. If the Extremities of the Proportionals, Σ, Δ, B, D, F, H, K, M, be joined by Right Lines, the Figure ΣΠ LM will become a Polygon consisting of more or less Sides, according as there are more or less Terms in the Progreffion.

If the Parts A C, CE, EG, GI, IL, be bisected in the Points c, e, g, i, l, and there be again raised the Perpendiculars c d, ef, gh, ik, Im, which are mean Proportionals between AB, CD; CD, EF; EF, GH; GH, IK; IK, LM; then there will arise a new Series of Proportionals, whose Terms, beginning from that which immediately follows Unity, are double of those in the first Series, and the Differences of the Terms are become less, and approach nearer to a Ratio of Equality than before. Likewife in this new Series, the Right Lines A L, AC, express the Distances of the Terms L M, CD, from Unity; viz. fince AL is ten Times greater than Ac, L M shall be the tenth Term of the Series from Unity: And because A e is three Times greater than A c, e f will be the third Term of the Series, if cd be the firft; and there shall be two mean Proportionals between A B and ef; and between AB and L M there will be nine mean Proportionals.

And if the Extremities of the faid Lines, viz. B, d, D, f, F, h, H, &c. be joined by Right Lines, there will be a new Polygon made, confifting of more, but shorter Sides than the last.

If, If, again, the Distances A c, C, Ce, e E, &c. be supposed to be bisected, and mean Proportionals between every two of the Terms be conceived to be put at those middle Distances; then there will arife another Series of Proportionals, containing double the Number of Terms from Unity than the former does; but the Difference of the Terms will be lefs; and if the Extremities of the Terms be joined, the Number of the Sides of the Polygon will be augmented according to the Number of Terms; and the Sides thereof will be leffer, because of the Diminution of the Diftances of the Terms from each other.

Now, in this new Series, the Distances AL, AC, &c. will determine the Orders or Places of the Terms; viz. if A L be five Times greater than A C, and CD be the fourth Term of the Series from Unity, then L M will be the twentieth Term from Unity.

If in this manner mean Proportionals be continually placed between every two Terms, the Number of Terms at last will be made so great, as also the Number of the Sides of the Polygon, as to be greater than any given Number, or to be infinite; and every Side of the Polygon so lessened, as to become less than any given Right Line; and confequently the Polygon will be changed into a curve-lined Figure; for any curveJined Figure may be conceived as a Polygon, whose Sides are infinitely small, and infinite in Number.

A Curve described after this manner is called Logarithmetical; in which, if Numbers be represented by Right Lines standing at Right Angles to the Axis AN, the Portion of the Axis intercepted between any Number and Unity thews the Place or Order, which that Number obtains in the Series of Geometrical Proportionals, diftant from each other by equal Intervals. For Example; if A L be five Times greater than A C, and there are a thousand Terms in continual Proportion, from Unity to LM; then will there be two hundred Terms of the fame Series from Unity to CD, or CD shall be the two hundredth Term of the Series from Unity; and let the Number of Terms from A B to L M be supposed what it will, then the Number of Terms from A B to CD will be one fifth Part of that Number,

The

The Logarithmetical Curve may also be conceived to be described by two Motions, one of which is equable, and the other accelerated, or retarded, according to a given Ratio. For Example, if the Right Line A B moves uniformly along the Line AN, so that the End A thereof describes equal Spaces in equal Times; and, in the mean Time, the faid Line A B so increases, that the Increments thereof, generated in equal Times, be proportional to the whole increasing Line, that is, if A B, in going forward to cd, be increased by the Increment od, and in an equal Time when it is come to CD, the Increment thereof is Dp, and Dp to do is as do is to AB; that is, if the Increments generated in equal Times are always proportional to the Wholes; or, if the Line A B, moving the contrary Way, diminishes in a conftant Ratio, fo that while it goes thro the equal Spaces, the Decrements АВ-ГД, ГАΠΣ, are proportional to A Β, ΓΔ; then the End of the Line, increasing or decreasing in the faid manner, describes the Logarithmetical Curve. For fince A B:do :: dc:Dp::DC:fq; it shall be (by Composition of Ratio), as A B: dc::dc:DC::DC:fe, and fo on.

By these two Motions, viz. the one equable, and the other proportionally accelerated or retarded, the Lord Neper laid down the Origin of Logarithms, and called the Logarithm of the Sine of any Arc, That Number which nearest defines a Line that equally increases, while, in the mean Time, the Line expreffing the whole Sine proportionally decreases to that Sine.

It is manifest, from this Description of the Logarithmetic Curve, that all Numbers at equal Distances are continually proportional. It is also plain, that if there be four Numbers A B, CD, IK, LM, fuch, that the Distance between the first and second be equal to the Distance between the third and the fourth: Let the Distance from the second to the third be what it will, these Numbers will be proportional. For, because the Distances AC, IL, are equal, A B shall be to the Increment Ds, as IK is to the Increment MT. Wherefore (by Composition) AB:DC:IK:ML. And contrariwise, if four Numbers be proportional, the Distance between the first and the second shall be be equal to the Distance between the third and the fourth.

The Distance between any two Numbers is called the Logarithm of the Ratio of those Numbers, and indeed doth not measure the Ratio itself, but the Number of Terms in a given Series of Geometrical Proportionals proceeding from one Number to another; and defines the Number of equal Ratios by the Compofition whereof the Ratios of Numbers are known.

If the Distance between any two Numbers be double to the Distance between two other Numbers, then the Ratio of the two former Numbers shall be the Duplicate of that Ratio of the two latter. For let the Distance I L between the Numbers IK, LM, be double to the Distance Ac, between the Numbers AB, cd; and fince I L is bisected in 1, we have A c=11,= IL; and the Ratio of IK tolm is equal to the Ratio of A B tocd; and so the Ratio of IK to LM, the Duplicate of the Ratio of 1K to l m (by Def. 10. El. 5.), shall be the Duplicate of the Ratio of A B to cd.

In like manner, if the Distance E L be triple of the Distance A C, then will the Ratio of EF to LM be triplicate of the Ratio of A B to CD: For, because the Distance is triple, there shall be three Times more Proportionals from EF to LM, than there are Terms of the fame Ratio from A B to CD; and the Ratio of EF to LM, as also of A B to CD, is compounded of the equal intermediate Ratios (by Def. 5. El. 6.) And so the Ratio of EF to LM, compounded of three Times a greater Number of Ratios, shall be triplicate of the Ratio of AB to CD. So, likewise, if the Distance G L be quadruple of the Distance Ac, then shall the Ratio of GH to LM be quadruplicate of the Ratio of A B to cd.

The Logarithm of any Number is the Logarithm of the Ratio of Unity to that Number; or it is the Distance between Unity and that Number. And fo Logarithms express the Power, Place, or Order, which every Number, in a Series of Geometrical Progrefsionals, obtains from Unity. For Example, if there be 10000000 proportional Numbers from Unity to the Number 10, that is, if the Number 10 be in the 10000000th Place from Unity; then it will be found

by

by Computation, that in the fame Series from Unity, to 2, there are 3010300 proportional Terms; that is, the Number 2 will stand in the 3010300th Place. In like manner, from Unity to 3, there will be found 4771213 proportional Terms, which Number defines the Place of the Number 3. The Numbers 10000000, 3010300, 4771213, shall be the Logarithms of the Numbers 10, 2, and 3.

If the first Term of the Series from Unity be called y, the second Term will be y2, the third y3, &c. And fince the Number 10 is the 10, 000, 000th Term of the Series, then will y۰۰۰۰۰۰۰ = 10; also y300300 = 2; alfo y 4771213 = 3; and fo on.

Wherefore all Numbers shall be some Powers of that Number which is the first from Unity; and the Indices of the Powers are the Logarithms of the Numbers.

Since Logarithms are the Distances of Numbers from Unity, as has been shewn, the Logarithm of Unity shall be 0; for Unity is not distant from itself: But the Logarithms of Fractions are negative, or descending below nothing; for they go on the contrary Way. And fo if Numbers, increasing proportionally from Unity, have positive Logarithms, or such as are affected with the Sign +, then Fractions or Numbers, in like manner decreasing, will have negative Logarithms, or such as are affected with the Sign-; which is true when Logarithms are confidered as the Diftances of Numbers from Unity.

But if Logarithms take their Beginning, not from an integral Unit, but from an Unit that is in some Place of decimal Fractions; for Example, from the Fraction 이이이이아이어어어어; then' all Fractions greater than this, will have positive Logarithms; and those that are lefs, will have negative Logarithms. But more shall be faid of this hereafter.

Since in the Numbers continually proportional, CD, EF, GH, IK, &c. the Distances CE, EG, GI, &c. are equal, the Logarithms AC, AE, AG, A I,

c. of those Numbers shall be equidifferent, or the Duferences of them shall be equal: And so the Logarithms of proportional Numbers are all in an Arithmetical Progreffion; and from hence proceeds that common Definition of Logarithms, viz. that Loga

« ZurückWeiter »