The Logarithmical 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 AB, moves uniformly along the Line AN, fo that the End thereof defcribes equal Spaces in equal Times; and, in the mean Time, the faid Line AB fo encreases, that the Increments thereof, generated in equal Times, be proportional to the whole encreafing Line; that is, if A B, in going forward to cd, be encreafed by the Increment od, and in an equal Time when it is come to CD, the Increment thereof is Dp, and Dp to de is as do is to AB, that is, if the Increments generated in equal Times are always proportional to the Wholes; or, the Line AB moving the contrary Way, diminishes in a conftant Ratio, fo that while it goes thro' the equal Spaces, the Decrements AB-ra гa, П, are Рroportionals to AB, гs. Then the End of the Line encreafing or decreasing in the faid Manner, describes the Logarithmical Curve: For fince AB: do:: dc:Dp::DC: fq, it fhall be (by Compofition of Ratio) as AB:de::de: 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 call'd the Logarithm of the Sine of any Arc, That Number which nearest defines a Line that equally encreafes, while, in the mean time, the Line expreffing the whole Sine proportionally decreases to that Sine.

It is manifeft from this Description of the Logarith mick Curve, that all Numbers at equal Diftances are continually proportional. It is alfo plain, that if there be four Numbers AB, CD, IK, LM, fuch, that the Dif tance between the first and fecond, 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 AC, IL, are equal, AB fhall be to the Increment Ds, as IK is to the Increment MT. Wherefore (by Compofition) AB: DC: IK: ML. And contrariwife, if four Numbers be proportional, the Distance between the firft and the fecond, fhall

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 Ratio's by the Compofition whereof the Ratio 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 the Ratio of the two latter. For let the Distance IL between the Numbers IK, LM, be double to the Distance Ac, between the Numbers AB, cd; and fince IL is bifected in 1, we have Ac =I=/L; 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 IK to Im, (by Def. 10. El. 5.) fhall be the Duplicate of the Ratio of AB to cd.

In like Manner, if the Distance EL be triple of the Distance A C, then will the Ratio of EF to LM, be triplicate of the Ratio of AB to CD: For because the Distance is triple, there fhall be three times more Proportionals from EF to LM, than there are Terms of the fame Ratio from AB to CD; and the Ratio of EF to LM, as alfo of AB to CD, is compounded of the equal intermediate Ratio's, (by Def. 5. El. 6.) And fo the Ratio of EF to LM, compounded of three times a greater Number of Ratio's, fhall 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 AB to cd.

!

The Logarithm of any Number is the Logarithm of the Ratio of Unity to that Number, or it is the Diftance 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 10, 2, and 3.

If the firft 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 10000010. Alfo y30103002. 이 Alfoy 73; and fo on.

Wherefore all Numbers fhall be fome Powers of that Number which is the firft 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 fhewn, the Logarithm of Unity fhall be o; for Unity is not distant 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 Sine; then Fractions or Numbers in like Manner decreafing, will have negative Logarithms, or fuch as are affected with the Sign-; which is true when Logarithms are confidered as the Distances of Numbers from Unity.

I

But if Logarithms take their Beginning not from an integral Unit, but from a Unit that is in fome Place of Decimal Fractions. For Example, from the Fraction 1000000000; 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, DC, EF, GH, IK, &c. the Diftances CE, EG, GI, &c. are equal, the Logarithms A C, AE, AG, AI, &c. of thofe Numbers fhall be equidifferent, or the Differences of them fhall be equal: And fo the Logarithms of proportional Numbers are all in an arithmetical Progreffion; and from hence proceeds that common Definition of Logarithms, that Loga

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

In the firft Kind of Logarithms that Neper publifh'd, the first Term of the continual Proportionals was placed only fo far diftant from Unity, as that Term exceeded Unity. For Example, if vn 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 1o fhall be the 23025850 Term of the Series, which Number therefore is the Logarithm of 10 in this Form of Logarithms, and expresses its Distance from Unity in fuch Parts whereof vy or An is one.

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But this Pofition is entirely at Pleasure; 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 fuppos'd at Pleafure,) that is between vy and By, the Increment of the first Term above Unity, and the Distance 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 1000,0000ta; and in this Form of Logarithms, the firft Increment vy fhall be to the Distance By, or An, as Unity, or AB, is to the Decimal Fraction 0,4342994, which therefore expreffes the Length of the Subtangent AT. 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 1 and 10, muft begin with o, or o muft ftand in the

firft Place to the Left-Hand; for they are leffer than the Logarithm of the Number 10, whofe Beginning is Unity; and the Logarithms of the Numbers between 10 and 100 begin with Unity; for they are greater than 1,0000000, and less than 2,0000000. Alfo the Logarithms between 100 and 1000, begin with 2, for they are greater than the Logarithm of 100, which begins with 2, and lefs than the Logarithm of a 1000 that begins with 3. In the fame Manner it is demonstrated, that the first Figure to the left Hand of the Logarithms between 1000 and 10000, muft be, 3; and the firft Figure to the left Hand of the Logarithms between 10000 and 100000, will be 4; and so on.

The firft Figure of every Logarithm to the left Hand, is called the Characteristick or Index, because it fhews the highest or moft remote Place of the Number from the Place of Units. For Example, if the Index of a Logarithm be 1, then the highest or most remote Place from Unity of the correspondent Number to the left Hand, will be the Place of Tens. If the Index be 2, the moff remote Figure of the correfpondent Number fhall be in the fecond Place from Unity; that is, it fhall be in the Place of Hundredths; and if the Index of a Logarithm be 3, the laft Figure of the Number anfwering to it, fhall be in the Place of Thousandths. The Logarithms of all Numbers that are in decuple or fubdecuple Progreffion, only differ in their Characteristicks, or Indices, they being written in all other Places with the fame Figures. For Example, the Logarithms of the Numbers 17, 170, 1700, 17000, are the fame, unless in their Indices; for fince is to 17, as 10 to 170, and as 100 to 1700, and as 1000 to 17000; therefore the Distances between 1 and 17, between 10 and 170, between 100 and 1700, and between 1000 and 17000, fhall be all equal. And fo fince the Distance between 1 and 17, or the Logarithm of the Number 17 is 1. 2304489, the Logarithm of the Number 170, will be 2.2304489, and the Logarithm of the Number 1700 fhall be 3.2304489, because the Logarithm of the Number 1002.0000000. In like Manner, fince the Loga rithm of the Number 1000 3.0000000, the Logarithm of the Number 17000 fhall 4. 2394489.,

So