Dexterity, compofed a Logarithmic Canon, agreeable to that new Form, for the first twenty Chiliads of Numbers (or from to to 20000), and for eleven other Chiliads, viz. from 90000 to For all which Numbers be calculated the Logarithms to fourteen Places of Figures. This Canon was published at London in the Year 1624.

1010000.

Adrian Vlacq published again this Canon at Gouda in Holland, in the Year 1628, with the intermediate Chiliads, before omitted, filled up according to Briggs's Prefcriptions; but thefe Tables are not fo ufeful as Briggs's, because the Loga-rithms are continued but to 10 Places of Figures.

Mr. Briggs has alfo calculated the Logarithms of the Sines and Tangents of every Degree, and the Hundredth Parts of Degrees to 15 Places of Figures; and has fubjoined 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. Thefe 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 feveral Places, compendious Tables, wherein the Sines and Tangents, and their Logarithms, confift of but seven Places of Figures, and wherein are only the Logarithms of the Numbers from 1 to 100000, which may be fufficient for moft Ufes.

The best Difpofition of thefe Tables, in my Opinion, is that first thought of by Nathanael Roe, of Suffolk; and with fome Alterations for the

Y 4

better,

better, followed by Sherwin in his Mathematical Tables, published at London in 1705; wherein are the Logarithms from 1 to 101000, confifting of feven Places of Figures. To which are fubjoined the Differences, and proportional Parts, by means of which, may be found easily the Logarithms of Numbers to 10000000; obferving, at the fame Time, that thefe Logarithms confift only of Seven Places of Figures. Here are alfo the Sines, Tangents, and Secants, with their Logarithm, and Differences for every Degree and Minute of the Quadrant, with fome other Tables of Ufe in pmactical Mathematics.

Of

A

S in Geometry the Magnitudes of the Lines are
often defined by Numbers; fo, likewise, on
the other hand, it is fometimes expedient to

expound Numbers by Lines, viz. by affuming fome Line which may reprefent Unity; the Double thereof, the Number 2; the Triple, 3; the one Half, the Fraction; and fo on. And thus the Genesis and Properties of fome certain Numbers are better conceived, and more clearly confidered, than can be done by abftract Numbers.

Hence, if any Line a* be drawn into itself, the Quantity a2, 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, likewife, if a2 be multiplied by a, the Product a3 will not be a Quantity of three Dimenfions, or a Geometrical Cube, but a Line that is the fourth Term in a Geometrical Progreffion, whofe firft Term is I, and fecond a; for the Terms 1, a, a2, a3, aa, a3, a“, a", &c. are in the continued Ratio of 1 to a. And the Indices affixed to the Terms fhew the Place or Distance that every Term is from Unity. For Ex

Fig. 1

ample,

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

If, between the Terms I and a, 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

I

Distance of a from Unity; and fo a may be written fora. And if a mean Proportional be put between a and a2, the Index thereof will be 1, or 2; for its Diltance will be fefquialteral of the 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 Unity;

and to the Cube Root may be expreffed by a. Hence the Index of Unity is o; for Unity is not diftant from itself.

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

a3, aa, a3, &c. are all in the fame Geometrical Progreffion. And fince the Distance of a from Unity is towards the Right Hand, and pofitive or + 1, the Distance equal to that on the contrary Side, viz. the

I

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

which fhall be the Index of the Term, for which

a

2

may be written a1. So likewife in the Terms a the Index-2 fhews, that the Term ftands in the second Place from Unity towards the Left Hand, and the Ex

preffions a 2 and are of the fame Value. Alfo a-3

is the fame as →→→.

a3

a2

For thefe negative Indices fhew,

that the Terms belonging to them go from Unity the

contrary

Contrary Way to that by which the Terms, whofe Indices are pofitive, do. Thefe Things premised,

If on the Line AN, both Ways indefinitely extended, be taken AC, CE, EG, GI, IL, on the Right Hand and alfo Аг,гП, &c. on the Left; all equal to one another; and if, at the Points II. I, A, C, E, G, I, L, be erected to the Right Line AN, the Perpendiculars П, TA, A B, CD, E F, GH, I K, L M, which let be continually proportional, and reprefent Numbers, whereof A B is Unity: The Lines A C, A E, AG, AI, AL,—Ar,—AП, refpectively exprefs 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 fhall be in the third Place from Unity, if CD be in the firft: So likewife fhall L M be in the fifth Place, fince AL=5AC. If the Extremities of the Proportionals, Σ, A, B, D, F, H, K, M be joined by Right Lines, the Figure ΣILM will become a Polygon confifting of more or lefs Sides, according as there are more or lefs Terms in the Progreffion.

If the Parts AC, CE, EG, GI, IL, be bifected in the Points c, e, g, 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 arife a new Series of Proportionals, whofe Terms, beginning from that which immediately follows Unity, are double of thofe in the first Series, and the Differences of the Terms are become lefs, and approach nearer to a Ratio of Equality than before. Likewife in this new Series, the Right Lines A L, A C, exprefs the Distances of the Terms LM, CD, from Unity; viz. fince A L is ten Times greater than A c, L M fhall be the tenth Term of the Series from Unity: And because A e is three Times greater than A c, ef will be the third Term of the Series, if c d be the firft; and there fhall be two mean Proportionals between A B and ef; and between A B and L M there will be nine mean Proportionals.

And if the Extremities of the faid Lines, viz. B, d, D, ƒ, F, h, H, &c. be joined by Right Lines, there will be a new Polygon made, confifting of more, but horter Sides than the laft.

If