The fame in Numbers; with the Operation.

As 13514224: 230: 10000000: 170.19, &c.

230

13514224) 2300000000 (170.19, &c. =AC. 13514224

The Analogy for BC.

As R AC seC BC. That is,

The fame in Numbers; with the Operation.

As 10000000: 170.191,&c.:: 16811730: 286.12,&c.

Thus is the First Cafe, in all its Varieties, refolved by the Method of Natural Numbers; and after this manner may all the other Cafes be refolved; there being nothing new or different therein from what is here done.

And, as I have hinted before, this Method may in fome Cafes be more preferable than the foregoing by Logarithms; for though that be moft expedite and eafy, yet this gives the Answer with greatest Exactnefs; especially where the Numbers of the given Triangle be large, and the Decimal Parts required to 4 or 5 Places, which fometimes does happen.

It must be observed alfo, that when Radius is the First Term in the Analogy, the Solution will be most exact, and the Operation more eafy; as may be feen in Scheme II and III.

CHA P. IX.

Of Solving Right-angled Plain Triangles by the Third Method, viz. by the Trigonometrical Sliding-Rule.

TH

HIS Third Method by the Sliding Rule is the moft ready and practical Way of any; and is for the most part pretty exact; though when the Quantity of the Angle (of which you use the Sine or Tangent) exceeds 40 or 50 Degrees; then great Exactncís must not be expected by the common fized Rule, which is a Foot long. But the longer the Rule is, the better or more exact and ufeful it will

be

be of Confequence; and where much Practice in Trigonometry happens, it may be worth while to have one made 4, 5, or 6 Feet long.

A Defcription of the Sliding Rule.

This Rule, as I have faid, is generally about 12 Inches in Length, and does confift of two Parts, viz. a fixed and a moveable Part or Slider; on each of which are certain graduated Lines, ferving for divers Purposes; As 1. A Line of Inch Measure on one of the Edges. 2. A Line of Equal Parts, marked at the Beginning EP. 3. Adjoined to that, a Line of Meridional Parts, for graduating Mercator's Chart, marked M. 4. A Line of Leagues, marked Leag. and, 5. A Line of Longitudes fitted thereto, marked M. Lon. 6. A Line of Rhumbs fitted thereto. 7. A Line of Chords, the one marked R. the other C. The four Lines laft mentioned, ferve to the Ufes of Navigation. 8. On the other Side the Rule, on one Extremity is another Line of Inches; and, 9. On the other Extremity, a Line of the fame Length, graduated into 100 equal Parts; by means of thele two Lines, is given by Infpection, the Decimal Parts anfwering to any Number of Inches and Parts of an Inch. 10. On one Side the Rule, on each Side the Grove, is placed a Line of Numbers, or Gunter's Line, marked N. 11. On the other Side the Rule, on one Side the Grove, is a Line of Sines, marked S. On the other Side the Grove, is a Line of Tangents, marked T. 13. On one of the Slider or moveable Piece, is a Line of Sines; and, 14. A Line of Tangents, both marked as before. 15. On the other Side the Slider, is a Line of Numbers; and, 16. A Line of Rumb-Sines; and thefe Lines I have now defcribed are all that are ufually put on the Sliding Rule, or Inftrument here spoken of.

12.

Of

Of all thofe Lines, the Lines of Numbers, Sines and Tangents, are the only ones ufed in folving a Plain Triangle; and because it will not a little conduce to form a right Notion of uling them, I fhall hint a Word or two concerning their Make and Conftruction: The Line of Numbers is Nothing but the Logarithms of the Natural Numbers, taken out of the Tables and laid on the Line, not regarding the Indices of the Logarithms. Thus for the greater Divifions of the faid Line, viz. from I to ro; the Logarithms to be taken from a Scale of equal Parts, for cach Divifion on the Line, ftand thus,

Divifions 2. 3. 4. 5. 6. 7. 8. 9. Logarithms .301 .477 .602 .698 778 .845 .903 954 For the leffer Divifions between 1 & 2, 2 & 3, &c. thus, Divifion 1.1. 1.2, &c. 2.1 2.2, &c. 3.1 3.2

Logarith. .041 .079,86..322.342, &c. .491.505586.

This Conftruction of the Line of Numbers being very well understood, as I prefume it eafily may; the Conftruction alfo of the Lines of Sines and Tangents on the Rule is thence fufficiently evident; for fince the Artificial Sines and Tangents are but the Logarithms of the Natural Numbers expreffing the fame Things, therefore it follows, that thofe Artifical Sines and Tangents, in the Manner before taught, may be laid down on the Scale, and there form the Lines we now fpeak of. And which I doubt not but the ingenious Young Trigonometer will efteem it only his Diverfion to do.

The Defcription, Nature and Conftruction of these Lines, being thus premifed, will make the Directions for their Ufe the fhorter, and Reason thereof most plain and obvious. From hence alfo it appears that both the preceeding Methods, are in Substance, the

fame with this, in a diverfe Manner apply'd: But whereas in them Exactness is the greatest Thing to be looked to, fo Eafe and Expedition are the chief Properties of this Third Method by the Sliding Rule.

One Thing I must not omit, and that is, to_acquaint the young Reader, that as there is no Line of Secants on the Rule, fo he must always obferve to frame fuch Proportions as admit of only Sines and Tangents, when he works by the Sliding Rule; for as there is feldom any Neceffity for ufing Secants, fo there is very rarely any Occafion for them; and when there is, I fhall fhew a Means to refolve it by the Rule nevertheless.

In order to perform Operations by the Sliding Rule, I fhall call that Line of Numbers on the Rule it felf A, but that Line on the Slider B; alfo I call the Line of Sines on the Rule F, and that on the Slider S. Alfo Note, That the End of each Line, viz. of Sines and Tangents, at 90° in the one, and at 45° in the other, is Radius in Analogies wrought this way.

Here alfo the foregoing Schemes and Analogies are to be used, and needs not either of them to be here again repeated; the manner of operating the first Cafe in each Scheme by the Line of Artificial Sines and Tangents now follow.

Cafe I. Scheme I. To find the Perpendicular AC.

Direction I. Caufe the Line of Sines to flide by each other, on one Side the Rule; then will the Lines of Numbers do fo on the other Side. II. Set 53° 30′ on S, to 36° 30' on F; then, III. Look on the other Side, and against 230 on B, is 170.19, &c. on A, the Length of AC, as before..

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