4

If, again, the Distances Ac, c C, Ce, e E, &c. be fuppofed to be bifected, and mean Proportionals between every two of the Terms be conceived to be put at thofe middle Distances; then there will arise 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, becaufe of the Diminution of the Dif tances of the Terms from each other.

Now, in this new Series, the Distances A L, AC, &c. wili determine the Orders or Places of the Terms; viz. if AL be five Times greater than AC, and CD be the fourth Term of the Series from Unity, then LM will be the 20th Term from Unity.

If in this manner mean Proportionals be continually placed between every two Terms, the Number of Terms at laft will be made fo great, as alfo 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 fo leffened, as to become lefs than any given Right Line; and confequently the Polygon will be changed into a curve-lined Figure; for any curvelined Figure may be conceived as a Polygon, whofe Sides are infinitely fmall, and infinite in Number.

A Curve defcribed after this manner is called Logarithmetical; in which, if Numbers be reprefented by Right Lines ftanding at Right Angles to the Axis AN, the Portion of the Axis intercepted between any Number and Unity fhews 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 B, and there are a thoufand Terms in continual Proportion from Unity to L M ; then there will be two hundred Terms of the fame Series from Unity to CD, or CD fhall be the two hundredth Term of the Series from Unity; and let the Number of Terms from A B to L M be fuppofed 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 defcribed 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 A N, fo that the End A thereof defcribes equal Spaces in equal Terms; and, in the mean Time, the faid Line AB to increases, that the Increments thereof, generated in equal Times, be proportional to the whole increafing Line, that is, if AB, 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 D p, and Dp to dc 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 through the equal Spaces, the Decrements A B-TA, TAП, are proportional to A B, TA; then the End of the Line increafing or decreasing in the fame manner, describes the Logarithmetical Curve. For fince AB: do ::dc: Dp::DC: fq; it fhall be (by Compofition of Ratio), as A B : de::dc: DC:: DC: ƒe, 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 increafes, while, in the mean Time the Line expreffing the whole Sine proportionally decreafes to that Sine.

It is manifeft, from this Defcription of the Logarithmetic Curve, that all Numbers at equal Diftances are continually proportional. It is alfo plain, that if there be four Numbers, A B, C D, IK, L M, 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 fecond to the third be what it will, thefe Numbers will be proportional. For, because the Distances A C, I L, are equal, A B fhall be to the Increment Ds, as I K is to the Increment M T. Wherefore, (by Compofition) AB: DC: IK: ML. And contrarywife, 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 thofe 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 fhall be the Duplicate of that Ratio of the two latter. For, let the Distance IL between the Numbers IK, LM, be double to the Distance A c, between the Numbers A B, ed; and fince I L is bifected in 7, we have A c=1 /=, IL; and the Ratio of IK to im is equal to the Ratio of AB to cd; and fo the Ratio of IK to LM, the Duplicate of the Ratio of I K to 1m (by Def. 10. El. 5.), íhall be the Duplicate of the Ratio of C B toc d. 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, becaufe. the Distance is triple, there fhall be three Times more Proportionals from E F to L M, than there are Terms of the fame Ratio from A B to CD; and the Ratio of EF to LM, as alfo of A B to CD, is compounded of the equal intermediate Ratios (by Def. 5. El. 6.) And fo the Ratio of E F to L M compounded of three Times a greater Number of Ratios, fhall be triplicate of the Ratio of A B to CD. So likewife, if the Distance GL be quadruple to the Distance A c, then fhall the Ratio of G H to L M be quadruplicate of the Ratio of A B to c d.

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 exprefs the Power, Place, or Order, which every Number, in a Series of Geometrical Progreffionals, 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 ftand 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, fhall be the Logarithms of the. Numbers 1, 2, and 3.

If the first Term of the Series from Unity be called y, the fecond Term will be y2, the third y3, &c. And fince the Number 10 is the 10, 000, 000th Term of the Series, then will y°°°°°°° = 10; alfo y301300 =2; alfo y47712 3; and fo on.

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

Since Logarithms are the Distances of Numbers from Unity, as has been fhewn, the Logarithm of Unity fhall be o; for Unity is not diftant from itself: But the Logarithms of Fractions are negative, or defcending below nothing; for they go on the contrary Way. And fo if Numbers, increafing proportionally from Unity, have pofitive Logarithms, or fuch as are affected with the Sign +, then Fractions or Numbers, in like manner decreafing, will have the negative Logarithms, or such as are affected with the Sign-; which is true when Logarithms are confidered as the Diftance of Numbers from Unity.

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

Since in the Numbers continually proportional, CD, EF, GH, IK, &c. the Distances CE, E G, GI, &c. are equal, the Logarithms AC, A E, A G, AI, &c. of thofe Numbers fhall be equidifferent, or the Differences of them fhall be equal: And fo the Lo garithms of proportional Numbers are all in an Arithmetical Progreffion; and from thence proceeds that common Definition of Logarithms, viz, that Loga

rithms are Numbers, which being adjoined to Proportions, have equal Differences.

In the firft Kind of Logarithms that Neper publifhed, the firft Term of the continual Proportionals was placed only fo far diftant from Unity, as that Term exceeded Unity. For Example, if v n be the firft Term of the Series from Unity A B, the Logarithm thereof, or the Distance An, or By, was according to him, equal to vy, or the Increment of the Number above Unity. As fuppofe vn be 1,0000001, he placed 0,0000001 for its Logarithm An; and from hence, by Computation, the Number 10 fhall be the 23025850th Term of the Series; which Number therefore is the Logarithm of 10 in this Form of Logarithms, and expreffes its Diftance from Unity in fuch Parts whereof vy or A n is one.

But this Pofition is entirely at Pleafure; for the Distance of the firft Term may have any given Ratio to the Excefs thereof above Unity; and according to that various Ratio (which may be fuppofed at Pleafure,) that is, between vy and By, the Increment of the firft Term above Unity, and the Diftance of the fame from Unity, there will be produced different Forms of Logarithms.

This firft Kind of Logarithms was afterwards changed by Neper, into another more convenient one, wherein he put the Number 10 not as the 23025850th Term of the Series but the 10000000; and in this Form of Logarithms, the first Increment vy fhall be to the Distance By, or An, as Unity, or A B, is to the Decimal Fraction 0,4342994, which therefore expreffes the Length of the Subtangent A T, Fig. 4.

After Neper's Death, the excellent Mr. Henry Briggs, by great Pains, made and published Tables of Logarithms according to this Form. Now fince in these Tables, the Logarithm of 10, or the Distance thereof from Unity, is 1,0000000, and 1,10,100, 1000, 10000, &c. are continual Proportionals, they fhall be equidiftant. Wherefore the Logarithm of the Number 100 fhall be 2,0000000; of 1000, 3,0000000; and the Logarithm of 10000 fhall be 4,0000000; and fo on. Hence the Logarithms of all Numbers between I and 10 muft begin with o, or o muft ftand in the