rather, Summary) of the Mathematicks, to have those 1everal Methods above-mention'd exemplified in fuch Cafes wherein each Method is more particularly neceffary; and this will be, to the apprehenfive Youth, as fufficient, as if every Cafe in every Triangle were treated in every particular Method it was capable of; which would of it felt make a large Volume.

In order that the Young Trigonometer may proceed with Clearnefs and Certainty, I fhall lay down the following Precepts, which he muft carefully obferve in his Operations, efpccially thofe which are wrote by Analogy.

Precept I. In forming you Analogies, you must always compare oppofite Sides to oppofite Angles; and

contra.

Precept II. When a Side is required, you must begin with an Angle; and when an Angle is fought, begin with a Side.

Precept III. When the Hypothenufe is given, you muft work with the Sine or Co-Sine according as the Side fought, is oppofite or adjacent to the given Angle. Precept IV. When the Hypothenufe is not known, you must work with Tangents and Co-Tangents, or with Secants; according as the Side is oppofite or adjacent to the Angle.

Precept V. Confider, that there is in the Tables of Sines, Tangents, &c. a Triangle exactly fimilar to the Triangle you are to folve; and whofe Sides expreffed in the Tables, are in the very fame Proportion as thofe of the Triangle propofed. Precept VI. Therefore (by Theorem XV.) you must fay, As the Length of any one Side, in Inches, Yards, Miles, &c. of the tabular Triangle is to a fimilar Side of the fame Measure in your Triangle; So is any other Side of the tabular Triangie, to the

fimilar

fimilar Side fought in your given Triangle; which Sides must be properly expreffed by Precept 3.4. Precept VII. When you are to use the Tables of Logarithms, if it be for the Logarithm of a Common Number, which in the Tables proceed from 1 to 1000, look in the Column under the Letter N, for the given Number,and right against it in the Column under the Word Logarith. you find its Logarithm: Thus, against the natural Namber 2165 you find its Logarithm, 3.3354579.

Precept VIII. If you are to find the Logarithm of any Sine, Tangent, or Secant of any Angle; feek, in the Canon of Artificial Sines, Tangents, &c. for the Degrees on the Top of the Page, and the Minutes in the firft Left-hand Column downward under the Letter M; if the Degrees be under 45°. thus against 25° 47' you find the Logarithm of the Sine (under the Word Log. Sinus) to be 9.6384585, and the Tangent (under the Word Log. Tang.) to be 9.6840011.

Precept IX. If your Degrees be more than 45°, you must seek the Degrees at the Bottom of the Page, and the Minutes in the Right-hand Column upwards; and thus against 57° 35′ you find the Sine (under Log. Sine) to be 9 9264310; and the Tangent (under Log. Tang.) to be 10.1972075. After the fame manner, the Natural Sines, Tangents, &c. are to be found.

Precept X. If the Table of Logarithmic Sines and Tangents contain alfo Logarithmic Secants, they are to be found as before directed for Sines and Tangents. But because few Tables do, you must, in Cafe they are wanting, proceed after this Manner; From double the Log. Radius, (which is always 20.0000000) fubtract the Log Co-Sine, the Log. Secant will remain. Example of 57° 35'•

Thus

Thus from the Double Radius

20.0000000

9.7292234 =10.2707766

Subtract the Log. Co-Sine of 57° 35′ There remains (by Theo. XVIII.) the Co-Secant thereof Precept XI. If you have a Logarithm, and would find the Number; or if it be the Logarithm of a Sine, Tangent, &c. and you would find the Degrees and Minutes anfwer thereto; proceed with the Logarithm to the Table, and take either the Number, or the Degrees and Minutes which ftand against the Logarithm next lefs than yours. Thus against the Logarithm next lefs to 2.8853912 I find the Number 768; and against the Logarithmic Sine next lefs than 9.9652379 I find 67° 22′. Note. The Places of Figures in the Number fought, must be always one more than the Index of the Logarithm.

Precept XII. In your Operations, if Radius be the fi.ft Term in your Proportions, you perform then by Addition only; which may alfo be done in any other Cafe; if inftead of the firft Term, you put its Arithmetical Complement (which is nothing but its Complement to 10.0000000) then add all the three Terms together, and the Sun (rejecting Radius) is the fourth Term which is fought. The Arithmetic Complement is moft eafily had, by mentally fubftracting each Figure of the Log. from Nine, and the laft from Ten; beginning from the Left-hand, thus the Arithmetic Compleinent of the Logarithm 4.5882892 is 5.41 17108, and 'tis most ingenious to work this way, as well as moft expeditious.

Thefe twelve Precepts must be well imprinted in the Memory of all who would be ready and dexterous in this moit excellent Art. Now to the Matter in Hand directly, viz. The Refolution of Triangles according to various Methods above-mentioned.

CHAP

CHA P. VII.

Of the First Method of Solving Rightangled Plain Triangles, by Artificial Sines, Tangents and Secants.

T will be fufficient to exemplify and illuftrate this and fome other Methods, in the Solution of the firft Cafe only; fince by making each Side Radius therein all the Variety of Proportions by Sines, Tangents and Secants, will come into Ufe, as is evident in the Synopfis.

And in order to make the whole Affair plain and evident to the young and untaught Faculties of Tyro's, I fhall proceed in an unufual Manner, and by a new kind of Scheme, whereby the Reafon of every Part of the Operation will be obvious, and easy to be understood

The Scheme is a Quadrant of a Circle, in which is difcribed a Triangle, confifting partly of black and partly of dotted Lines, viz. aBc, the very fina!! Part thereof terminated by black Lines, viz. ABC, reprefents the Triangle given to be refolv'd, and is like or fimilar to firft Great One, which may be call'd the Original, or Tabular Triangle; because all the dotted Part reprefents that which is contain'd in the Tables. And because every Side of the Original or Tabular Triangle is known in the Tables, and the given One in every Part fimilar thereto, therefore the Quafita of the propofed Triangle is found by fuch Analogy or Proportion; as per Theor. XV. and Precept V.

The

The Sides of the Original Triangles are computed in fuch Parts as the Radius confifts of 10000000000, the Logarithm of which Number 10.0000000, as in the Table. Accordingly the other Sides, as they are Sines or Tangents, confifts of lefs or more of thofe Parts; and the Judicies of their Logarithms are less or greater likewife.

But notwithstanding the Sides of the Original Triangle are calculated to Eight, Nine, Ten, &c. Places of Figures, for which Number of Places alfo the Logarithms are made and fitted; yet because in working Proportions the Truth may be attained to, with a fufficient Exactnefs, by a leffer Number of Figures, therefore the Natural Numbers in the Tables exceed not Seven or Eight Places in general.

And therefore (for Example) when in the Artificial Canon, you fee the Log. Sine 36° 30' to be 9.7743876; whofe Index being 9, fhews its Natural Number to confift of Ten Places; but yet in the Natural Canon, you find only the Number 5948228 to exprefs that Sine, which Number hath but Seven Places; therefore the three dificient Places muft be fupplied by Cyphers, and then the Number will be 5948228000, compleat in Number of Places, though detective in its Figures or Value, yet fufficient for

Use

Thefe Things premifed difplay the whole Nature and Miftery of Trigonometrical Calculation; to which I now proceed.

Cafe