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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 shall 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 l, we have Ac =I1=IL; and the Ratio of I K to lm, is equal to the Ratio of AB to cd; and so the Ratio of I K to LM, the Duplicate of the Ratio of IK to lm, (by Def. 10. El. 5.) shall 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 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 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 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, 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 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 express the Power, Place, or Order which every Number, in a Series of Geometrical Progressionals, obtains from Unity. For Example, if there be 1900gpoo proportional Numbers from Unity to the Number 10, that is, if the Number 1o be in the I cooooooth Place from Unity ; then it will be found,
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 3010300 Place. In like Manner, from Unity to 3, there will be found 4771213, proportional Terms, which Number defines the Place of the Number 3.1 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 y, the third y}, &c. And since the Number 1o is the 10,000,000th Term of the Series, then will yoðJOÓS=10. | Also 93010300=2., Allo y :7111 =3; and so on.
Wherefore all Numbers shall be some 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 shewn, the Logarithm of Unity shall be o; 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 Sinet; 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 considered as the Distances of Numbers from Unity.
But if Logarithms take their Beginning not from an integral Unit, but from a Unit that is in some Place of Decimal Fractions. For Example, from the Fraction 10005000000; then all Fractions greater than this, will have positive Logarithms, and those that are less, will have negative Logarithms. But more fhall be faid of this hereafter.
Since in the Numbers continually proportional, DC, EF, GH, IK, &c. the Distances CE, EG, GI, &c. are equal, the Logarithms A C, AE, AG, AI, &c. of those Numbers shall 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 hence proceeds that common Definition of Logarithms, that Loga
rithms are Numbers which, being adjoined to Proportions, have equal Differenccs.
In the first 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 un 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 suppose un be 1,0000001, he placed 0,0000001 for its Logarithm An; and from hence, by Computation, the Number ro 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.
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 fuppos'd at Pleafure,) that is between vy and By, the Increment of the first Term above Unity, and the Distance of the same 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 23025850h "Term of the Series, but the 1000,00oot; 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 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 since 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 equidistant. Wherefore the Logarithm of the Number 100 shall be 2,0000000 ; of 1000, 3,0000000"; and the Logarithm of 10000 shall be 4,0000000; and so on.
Hence the Logarithms of all Numbers between 1 and 10, must begin with o, or o must stand in the 3
first Place to the Left-Hand; for they are leffer than the Logarithm of the Number 10, whose 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. Also the Logarithms between 100 and 1000, begin with 2, for they are greater than the Logarithm of 100, which begins with 2, and less than the Logarithm of a 1000 that begins with 3. In the same Manner it is demonstrated, that the first Figure to the left Hand of the Logarithms between 1ooo and 10000, must be, 3 ; and the first Figure to the left Hand of the Logarithms between
een 10000 and 100000, will be 4; and so on.
The first Figure of every Logarithm to the left Hand, is called the Characteristick or Index, because it fhews the highest or most 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 most fremote Figure of the correspondent Number shall be in the second place from Unity; that is, it shall be in the Place of Hundredths ; and if the Index of a Logarithm be 3, the last Figure of the Number answering to it, shall be in the Place of Thousandths. The Logarithms of all Numbers that are in decuple or fubdecuple Progression, only differ in their Characteristicks, or Indices, they being written in all other Places with the same Figures. For Example, the Logarithms of the Numbers 17, 170, 1700, 17000, are the fame, unless in their Indices; for fince 1. is to 17, as 10 to 170, and as 100 to 17001 and as 1000 to 17000 ; therefore the Distances between 1 and 17, between 10 and 170, between 190 and 1700, and between 1000 and 17000, shall 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 shall be 3. 2304489, because the Logarithm of the Number 10032.0000000. In like Manner, since the Logarithm of the Number 1000=3.0000000, the Logarithm of the Number 17000 shall 4. 2394489.
So also the Numbers, 6748. 674, 8. 67, 48.6, 748. 0, 6748. o, 06748, are continual Proportionals in the Ratio of 10 to 1 ; and so their Distances from each 6 7 4 8 3,8291751 other shall be equal to the 67 4,8 2,8291751 Distance or Logarithm of 6 7, 48 1,8291751 the Number 10, or equal 6, 7 4 8 0,8291751 to 1,0000000. And so fince 0,6 7 4 8-1,8292751 the Logarithm of the Num- 0,06 7481-2,8291751 ber 6748 is 3,8291751, the Logarithms of the other Numbers fhall be as in the Margin ; where you may observe that the Indices of the last two Logarithms are only negative, and the other Figures positive ; and so when those other Figures are to be added, the Indices must be fubtracted, and contrariwise.
CH A P. II.
Of the Arithmetick of Logarithms in rehole
Numbers, or whole. Numbers adjoined to decimal Fractions. Fig. 2:
Ecause, in Multiplication, Unity is to the Mukiplier, as the Multiplicand is to the Product, the
Distance between Unity and the Multiplier, thall be equal to the Distance between the Multiplicand and the Product ; if therefore, the Number G H be to be multiplied by the Number EF, the Distance between GH and the Product must be equal to the Diftance A E, or to the Logarithm of the Multiplier ; and to if G L be taken equal to AE, the Number L M fhall be the Product, that is, if the Logarithm of the Multiplicand AG be added, to the Logarithm of the Multiplier A E, the Sum fhall be the Logarithm of the Product.
In Division, the Divisor is to Unity, as the Divi dend is to the Quotient; and so the Distance between the Divisor and Unity shall be equal to the Diftance between the Dividend and the Quotient. So if L M be to be divided by EF, the Distance EA hall be equal to the Diftance between L M and the Quotient,