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If, again, the Distances Ac, C, C, e E, &c. be supposed to be bisected, and mean Proportionals between every two of the Terms be conceived to be put at thoe 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 less.; and if the Extremities of the Terms be joined, the Number of the sides of the Polygon will be augmented according to the Numbes of Terms ; and the Sides thereof will be lefser, because of the Diminution of the Diftances 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 2011 Term from Unity.
If in this manner mean Proportionals be continuo ally 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 leftened, as to become less 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, 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 fhews the Place or Order, which that Number obtains in the Series of Geometrical Proportionals, distant from each other by equal Intervals. For Example, if A L be five Times greater than A B, and there are a thousand Terms in continual Proportion from Unity to L M ; then there will be two hundred Terms of the same 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 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 AB moves uniformly along the Line AN, so that the End A thereof describes equal Spaces in equal Terms; and, in the mean Time, the said Line AB to increases, that the Increments thereof, generated in equal Times, be p'oportional to the whole increasing 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 D p to do is as do is to AB; that is, if the Increments generated in equal Tiines are always proportional to the Wholes; or, if the Line A B, moving the contrary Way, diminishes in a constant Ratio, so that while it goes through the equal Spaces, the Decrements A B-ra, raNE, are proportional to AB, TA; then the End of the Line increasing or decreasing in the fame manner, describes the Logarithmetical Cusve. For fince AB: d. ::dc:Dp::DC:fq; it shall be (by Composition of Ratio), as A B :dc::dc:DC::DC:fe, and
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 neareft defines a Line that equally increases, while, in the mean Time the Line exprefing 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, L M, such 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 A C, I L, are equal, A B shall be to the Increment Ds, as I K is to the Increment MT. Wherefore, (by Compofition) AB: DC:IK:ML. And contrarywife, if four Numbers be proportional, the Distance between the first and the second fall
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 Composition whereof the Ratios of Numbers are known.
If the Distance between any two Numbira be două ble 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 I L between the Numbers IK, LM, be double to the Distance A c, between the Numbers A B, cd; and fince I L is bisected in l, we have A c=1 l, IL; and the Ratio of IK to lmis equal to the Ratio of AB to cd; and to the Ratio of I K to LM, the Duplicate of the Ratio of I K to 1m (by Def. 10. El. 5.), ihall be the Duplicate of the Ratio of C B toc d.
In like manner, if the Diftance E L be triple of the Distance A C, then will the Ratio of E F 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 E F to L M, than there are Terms of the samne Ratio from A B to CD; and the Ratio of E F to L M, as also of A B to CD, is compounded of the equal intermediate Ratios (by Def. 5. El. 6.) And so the Ratio of E F to L M compounded of three Times a greater Number of Ratios, shall be triplicate of the Ratio of A B to CD. So likewise, if the Distance G L be quadruple to the Distance A c, then shall the Ratio of G H to L M be quadruplicate of the Ratio of A B toc 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 so Logarithms express the Power, Place, or Order, which every Number, in a series of Geometrical Progresfionals, obtains from Unity. For Example, if there be 10000000 proportional Numbers from Unity to the Number 10, that is, if the Number 1o be in the 100000ooth Place from Unity, then it will be found
by Computation, that in the fame Series from Unity, to 2, there are 30 10300 proportional Terms; that is, the Number 2 will stand in the 30103ooth 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. 1, 2, and 3.
If the first Term of the Series from Unity be called
the second Term will be y?, the third y3, & c. And since the Number ro is the 10,000, oooth Term of the Series, then will y"0000000 = 10; also y3010360 =2 , also y 4.771213 = 3; and so on.
Wherefore all Numbers shall 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 thewn, the Logarithm of Unity shall be 0; for Unity is not diftant from itself: But the Logarithms of Fractions are negative, or de-. scending below nothing ; for they go on the contrary Way, And so 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 the negative Logarithms, or such as are affected with the Sign-; which is true when Logarithms are considered 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 some Place of decimal Fractions ; for Example, from the Fraction
Toto 600000; then all Fractions greater than this, will have positive Logarithms, and those that are less, will have negative Logarithms. But more shall be laid of this hereafter,
Since in the Numbers continually proportional, CD, EF, GH, IK, &, the Distances CE, EG, GI, &c. are equal, the Logarithms A C, A E, AG, AI, &c. of those Numbers Thall be equidifferent, or the Differences of them shall be equal : And so the Logarithms of proportional Numbers are all in an Arithmetical Progression ; and from thence proceeds that common Definition of Logarithms, viz, that Loga
rithms are Numbers, which being adjoined to Proportionis, have equal Differences.
In the first Kind of Logarithms that Neper pubJished, the first Term of the continual Proportionals was placed only so far distant from Unity, as that Term exceeded Unity. For Example, if u n be the first Term of the Series from Unity AB, the Logarithm thereof, or the Distance An, or B y, was according to him, equal to vy, or the Increment of the Number above Unity. As suppose un be 1,0000001, he placed 0,0000001 for its Logarithm An; and from hence, by Computation, the Number 10 Thall be the 23025850th Term of the Series ; which Number therefore is the Logarithm of 10 in this Form of Logarithms, and expreises its Distance from Unity in fuch Parts whereof vy or Anis one.
But this position is entirely at Pleasure ; for the Distance of the first Term may have any given Ratio to the Excess thereof above Unity; and according to that various Ratio (which may be supposed at Pleafure,) that is, between vy and B y, 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 first Kind of Logarithms was ' afterwards changed by Neper, into another more convenient one, wherein he put the Number 10 not as the 23025850th Terin of the Series but the 1000000011; and in this Form of Logarithms, the first Increment vy shall be to the Distance By, or An, as Unity, or A B, is to the Decimal Fraction 0,4342994, which therefore expresses 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 shall be equidiftant. Wherefore the Logarithm of the Number 100 shall be 2,0000000; of 1000, 3,0000000 ; and the Logarithm of 10000 Mall be 4,0000000 ; and so on.
Hence the Logarithms of all Numbers between I and 10 must begin with o, or o must stand in the