ively, the sum of the two rectangles will be equal to the square of the sum of those numbers.

Thus, 10+6= 16; now, 16 x 10 = 160; 16×6=96; and 160+96 = 256.

Again, 10+6= 16; and 16× 16 = 256.

The square of the sum of any two numbers is equal to four times the square of half their sum.-Thus,

=

10+6= 16; and 16 × 16 = 256; then 10+6 16+2 = S, and Sx 8 x4 = 256.

The sum of the squares of any two numbers is equal to the square of their difference, together with twice the rectangle of those numbers.Thus,

10×10 = 100; 6×6= 36; and 100+36= 136.-Again,

10-64; and 4 x 4 = 16; 10×6×2 120; and 120+16= 136. The numbers 3, 4 and 5, or their multiples 6, 8 and 10, &c. &c., will express the three sides of a right angled plane triangle.

The sum of any two square numbers whatever, their difference, and twice the product of their roots, will also express the three sides of a right angled plane triangle. Thus,

=

Let 9 and 49 be the two square numbers :-then 9+49 58; 49-9= 40.-Now, the root of 9 is 3, and that of 49 is 7;-then 7 × 3 × 2 = 42 : hence the three sides of the right angled plane triangle will be 58, 40, and 42.

The sum of the squares of the base and perpendicular of a right angled plane triangle, is equal to the square of the hypothenuse.

The difference of the squares of the hypothenuse and one leg of a right angled plane triangle, is equal to the square of the other leg.

The rectangle or product of the sum and difference of the hypothenuse and one leg of a right angled plane triangle, is equal to the square of the other leg.

The cube of any number divided by 6 will leave the same remainder as the number itself when divided by 6.-The difference between any number and its cube will divide by 6, and leave no remainder.

Any even square number will divide by 4, and leave no remainder; but an uneven square number divided by 4 will leave 1 for a remainder.

PLANE TRIGONOMETRY.

The Resolution of the different Problems, or Cases, in Plane Trigonometry, by Logarithms.

ALTHOUGH it is not the author's intention (as has been already observed,) to enter into the elementary parts of the sciences on which he may have occasion to touch in elucidating a few of the many important purposes to which these Tables may be applied; yet, since this work may, probably, fall into the hands of persons not very conversant with trigonometrical subjects, he therefore thinks it right briefly to set forth such definitions, &c. as appear to be indispensably necessary towards giving such persons some little insight into this particular department of science.

PLANE TRIGONOMETRY is that branch of the mathematics which teaches how to find the measures of the unknown sides and angles of plane triangles from some that are already known.-It is divided into two parts; right angled and oblique angled :—in the former case one of the angles is a right angle, or 90; in the latter they are all oblique.

Every plane triangle consists of six parts; viz., three sides and three angles; any three of which being given (except the three angles), the other three may be readily found by logarithmical calculation.

In every triangle the greatest side is opposite to the greatest angle; and, vice versa, the greatest angle opposite to the greatest side.-But, equal sides are subtended by equal angles, and conversely.

The three angles of every plane triangle are, together, equal to two right angles, or 180 degrees.

If one angle of a plane triangle be obtuse, or more than 90°, the other two are acute, or each less than that quantity: and if one angle be right, or 90, the other two taken together, make 90: :—hence, if one of the angles of a right angled triangle be known, the other is found by subtracting the known one from 90.-If one angle of any plane triangle be known, the sum of the other two is found by subtracting that which is given from 180; and if two of the angles be known, the third is found by subtracting their sum from 180:

The complement of an angle is what it wants of 90°; and the supplement of an angle is what it wants of 180:

In every right angled triangle, the side subtending the right angle is called the hypothenuse; the lower or horizontal side is called the base, and that which stands upright, the perpendicular.

If the hypothenuse be assumed equal to the radius, the sides, that is, the base and the perpendicular, will be the sines of their opposite angles. And, if either of the sides be considered as the radius, the other side will be the tangent of its opposite angle, and the hypothenuse the secant of the same angle.

Thus.-Let ABC be a right angled plane triangle; if the hypothenuse AC be made radius, the side BC will be the sine of the angle A, and AB the sine of the angle C.-If the side A B be made radius, BC will be the tangent, and AC the secant, of the angle A:-And, if BC be the radius, AB will be the tangent, and AC the secant of the angle C.

For, if we make the hypothenuse AC radius (Fig. 1.), and upon A, as a centre, describe the arch CD to meet AB produced to D; then it is evident that B C is the sine of the arch DC, which is the measure of the angle BAC; and that AB is the co-sine of the same arch :—and if the arch AE be described about the centre C, to meet C B produced to E, then will A B be the sine of the arch A E, or the sine of the angle AC B, and B C its co-sine.

Again, with the extent A B as a radius (Fig. 2.), describe the circle BD; then BC is the tangent of the arch B D, which is evidently the measure of the angle BAC; and AC is the secant of the same arch, or angle.

Lastly, with CB as a radius (Fig. 3.), describe

Sevant of C

A

גיין
דיועט

Roulins

Sine

Radias

Fig 1.

E

Fig: 3.

Fig 2

In the computation of right angled triangles, any side, whether given or required, may be made radius to find a side; but a given side must be made radius to find an angle: thus,

To find a Side:

Call any one of the sides of the triangle radius, and write upon it the word radius:-observe whether the other sides become sines, tangents, or secants, and write these words on them accordingly, as in the three preceding figures: then say, as the name of the given side, is to the given side; so is the name of the side required, to the side required.

And, to find an Angle :

Call one of the given sides the radius, and write upon it the word radius: observe whether the other sides become sines, tangents, or secants, and write these words on them accordingly, as in the three foregoing figures; then say, as the side made radius, is to radius; so is the other given side to its name: that is, to the sine, tangent, or secant by it represented.

Now, since in plane trigonometry the sides of a triangle may be considered, without much impropriety, as being in a direct ratio to the sines of their opposite angles, and conversely; the proportion may, therefore, be stated agreeably to the established principles of the Rule of Three Direct, by saying

As the name of a given angle, is to its opposite given side; so is the name of any other given angle to its opposite side.-And, as a given side, is to the name of its opposite given angle; so is any other given side to the name of its opposite angle.

The proportion, thus stated, is to be worked by logarithms, in the following manner; viz.,

To the arithmetical complement of the first term, add the logs. of the-second and third terms, and the sum (rejecting 20, or 10 from the index, according as the required term may be a side or an angle,) will be the logarithm of the required, or fourth term.

Remarks.-1. The arithmetical complement of a logarithm is what that logarithm wants of the radius of the Table; viz., what it is short of 10.000000; and the arithmetical complement of a log. sine, tangent, or secant, is what such logarithmic sine, &c. &c. wants of twice the radius of the Tables, viz., 20. 000000.

2. The arithmetical complement of a log. is most readily found by beginning at the left hand and subtracting each figure from 9 except the last significant one, which is to be taken from 10, as thus ;-if the given log. be 2.376843, its arithmetical complement will be 7.623157:-if a given log. sine be 9. 476284, its arithmetical complement will be 10. 523716, and

so on.

3. The arithmetical complement of the log. sine of an arch, is the log. co-secant of that arch ;-the arithmetical complement of the log. tangent of an arch, is the log. co-tangent of that arch; and conversely, in both

cases.

Solution of Right-angled Plane Triangles, by Logarithms.

PROBLEM I. .

Given the Angles and the Hypothenuse, to find the Base and the
Perpendicular.

Example.

Let the hypothenuse A C, of the annexed triangle A B C, be 246. 5, and the angle A 53:7:48"; required the base A B, and the perpendicular B C ?

Note. Since there is no more intended, in this place, than merely to show the use of the Tables; the geometrical construction of the diagrams is, therefore, purposely omitted.

By making the hypothenuse AC radius; BC becomes the sine of the angle A, and A B the co-sine of the same angle.-Hence,

Making the base AB radius; B C becomes the tangent of the angle A, and AC the secant of the same angle.-Hence,