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Dexterity, composed a Logarithmic Canon, agreeable to that new Form, for the first twenty Chiliads of Numbers (or from to i 10. 20000), and for eleven orber Chiliads, viz. from 90000 to 1010000.

For all wbicb Numbers be calculated the Logarithms to fourteen Places of Figures. This Canon was published at London in the Year 1624.

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Adrian Vlacq published again this Canon at Gouda in Holland, in the l'ear 1628, with the intermediate Cbiliads, before omitted, filled up according to Briggs's Prescriptions ; but ihese Tables are not so useful as Briggs's, because the Logarithms are continued but to 10 Places of Figures.

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Mr. Briggs has also calculated the Logarithms of the Sines and Tangents of every Degree, and the Hundredth Parts of Degrees to 15 Places of Figures ; and has subjoined to them the natural Sines, Tangents, and Secants, to 15 Places of Figures. The Logarithms of the Sines and Tangents are called artificial Sines and Tangents. These Tables, together with their Construction and Uje, were published after Briggs's Death at London, in the Year 1633, by Henry Gillebrand, and by him called Trigonometria Britannica.

Since then, there have been published, in several Places, compendious Tables, wherein the Sines and Tangents, and their Logarithms, consist of but seven Places of Figures, and wherein are only the Logaritbins of the Numbers from 1 10 100000, whicb inay be sufficient for most Uses.

The best Disposition of these Tables, in my Opinion, is that first thought of by Nathanael Roe, of Suffolk ; and with some . Alterations for the

better,

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better, followed by Sherwin in bis Mathematical Tables, published at London in 1705; wherein are the Logarithms from i to 101000, consisting of seven Places of Figures. To which are subjoined the Differences, and proportional Parts, by means of which, may be found easily the Logarithms of Numbers to 10000000 ; observing, al the same Time, that these Logarithms consist only of seven Places of Figures. Here are also the Sines, Tangents, and Secants, with their Logarithm, and Differences for every Degree oond Minute of the Quadrant, with some other Tables of Use in pmacdical Marbematics.

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OF THE

Nature and Arithmetic

OF

LOGARITHMS.

CH A P. I.

Of the ORIGIN and NATURE of

L O G A R I T H M S.

A

Fig. Iq

Sin Geometry the Magnitudes of the Lines are
often defined by Numbers ; lo, likewise, on
the other hand, it is sometimes expedient to

expound Numbers by Lines, viz. by afluming some Line which may represent Unity; the Double thereof, the Number 2 ; the Triple; 3 ; the one Half, the Fraction ; and so on. And thus the Genesis and Properties of some certain Numbers are better conceived, and more clearly considered, than can be done by abstract Numbers.

Hence, if any Line a* be drawn into itself, the Quantity a, produced thereby, is not to be taken as one of two Dimensions, or as a Geometrical Square, whose Side is the Line a, but as a Line that is a third Proportional to fome Line taken for Unity, and the Line 4. So, likewise, if a? be multiplied by a, the Product ay will not be a Quantity of three Dimensions, or a Geometrical Cube, but a Line that is the fourth Term in a Geometrical Progression, whose first Term is 1, and second a; for the Terms i, a, a, a3, 84, as, a', a', &c. are in the continued Ratio of 1 to a. And the Indices affixed to the Terms shew the Place or Distance that every Term is from Unity. For Ex

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ample, as is in the fifth Place from Unity, as in the fixth, or fix Times more distant from Unity than a, or a', which immediately follows Unity.

If, between the Terms i anda, there be put a mean Proportional, which is a, the Index of this will be for its Distance from Unity will be one half of the

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Distance of a from Unity; and so a may bę written for v a. And if a mean Proportional be put between a and a>, the Index thereof will be i į or ; for its Diltance will be sesquialteralofthe Distance of a from Unity.

If there be two mean Proportionals put between i and a; the first of them is the Cube Root of a, whose Index must be ; for that Term is diftant from Unity only by a third Part of the Distance of a from L'nity; and to the Cube Root may be expressed by a. Hence the Index of Unity is o; for Unity is not distant from itself.

The fame Series of Quantities, geometrically proportional, may be both ways continued, as well defeending toward the Left Hand, as ascending towards

II I II the Right; for the Terms-,, , ,

, I, a, a,

as at az a2 23, a4, as, &c. are all in the fame Geometrical Progression. And since the Distance of a from Unity is towards the Right Hand, and positive or + 1, the Distance equal to that on the contrary Side, viz. the

I

Distance of the Term-, will be negative, or -1,

I which shall be the Index of the Term for which

a

may

be written ai. So likewise in the Terms a the Index-2 shews, that the Term stands in the second Place from Unity towards the Left Hand, and the Ex

I preficns a 2 and—are of the same Value. Also a-3

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is the same as

For these negative Indices shew,

23

that the Terms belonging to them go from Unity the

contrary

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 AC, CE, EG, GI, IL, on the Right Hand i and also AT,III, &c. on the Left; all equal to one another, and if, at the Points II C, A, C, E, G,I, L, be erected to the Right Line AN, the Perpendiculars IIx, FA, A B C D E F G H I K L M, which let be continually proportional, and represent Numbers, whereof A B is Unity: The Lines A C, A E, A,G, AI, AL, AP,-AII, 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 C D be in the first : So likewise shall L M be in the fifih Place, fince AL=5AC. If the Extremities of the Proportionals, E, A, B, D, F, H, K, M be joined by Right Lines, the Figure II LM will become a Polygon consisting of more or less Sides, according as there are more or less Terms in the Progression.

If the Parts AC, CE, EG, GI,IL, be bisected in the Points C, e, f, i, l, and there be again raised the Perpendiculars c, d, e, f, g, h, i, k, l, m, which are mean Proportionals between A B, 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. Likewise in this new Series, the Right Lines A L, A C, express the Distances of the Terms L M, CD, from Unity ; viz. since AL is ten Times greater than Ac, L M fhall be the tenth Term of the Series from Unity : And because A e is three Times greater than Ac, ef will be the third Term of the Series, if c d be the fiift; and there shall be two mean Proporcionals between A Band e f ; and between A B and L M there will be nine mean Proportionals.

And if the Extremities of the said Lines, viz. B, ., D, f, F, b, H, &c. be joined by Right Lines, these will be a new Polygon made, consisting of incre, but borter Sides than the last,

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