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we have, in the inverted series
n=Ax+Bx2+Cx3+Dx1+Ex3 +, &c.

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According to the scheme lately introduced into France, of dividing the denominations of weights, measures, &c. into tenths, hundredths, &c. the fourth part of a circle is divided into 100 degrees, a degree into 100 minutes, a minute into 100 seconds, &c. The whole circle contains 400 of these degrees; a plane triangle 200. If a right angle be taken for the measuring unit; degrees, minutes and seconds, may be written as decimal fractions. Thus 36° 5′ 49′′ is 0.360549.

According to the French division


10° 9° 100' 54' 1000" 324"


NOTE F. p. 82.

If the perpendicular be drawn from the angle opposite the longest side, it will always fall within the triangle; because the other two angles must, of course, be acute. But if one of the angles at the base be obtuse, the perpendicular will fall without the triangle, as CP, (Fig. 38.)

In this case, the side on which the perpendicular falls, is to the sum of the other two; as the difference of the latter, to the sum of the segments made by the perpendicular.

The demonstration is the same, as in the other case, except that AH-BP+PA, instead of BP-PA.

Thus in the circle BDHL (Fig. 38.) of which C is the center,

ABXAH ALX AD; therefore AB: AD::AL:: AH.




When the three sides are given, it may be known whether one of the angles is obtuse. For any angle of a triangle is obtuse or acute, according as the square of the side subtending the angle is greater, or less, than the sum of the squares of the sides containing the angle. (Euc. 12, 13. 2.)

NOTE G. p. 104.

Gunter's Sliding Rule, is constructed upon the same principle as his scale, with the addition of a slider, which is so contrived as to answer the purpose of a pair of compasses, in working proportions, multiplying, dividing, &c. The lines on the fixed part are the same as on the scale. The slider contains two lines of numbers, a line of logarithmic sines, and a line of logarithmic tangents.

To multiply by this, bring 1 on the slider, against one of the factors on the fixed part; and against the other factor on the slider, will be the product on the fixed part. To divide, bring the divisor on the slider, against the dividend on the fixed part; and against 1 on the slider, will be the quotient on the fixed part. To work a proportion, bring the first term on the slider, against one of the middle terms on the fixed part; and against the other middle term on the slider, will be the fourth term on the fixed part. Or the first term may be taken on the fixed part; and then the fourth term will be found on the slider.

Another instrument frequently used in trigonometrical constructions, is


This consists of two equal scales movable about a point as a center. The lines which are drawn on it are of two kinds; some being parallel to the sides of the instrument, and others diverging from the central point, like the radii of a circle. The latter are called the double lines, as each is repeated upon the two scales. The single lines are of the same nature, and have the same use, as those which are put upon the common scale; as the lines of equal parts, of chords, of latitude, &c. on one face; and the logarithmic lines of numbers, of sines, and of tangents, on the other.

The double lines are

A line of Lines, or equal parts, marked Lin. or L.
A line of Chords,

Cho. or C.

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The double lines of chords, of sines, and of tangents to 45°, are all of the same radius; beginning at the central point, and terminating near the other extremity of each scale; the chords at 60°, the sines at 90°, and the tangents at 45°. (See Art. 95.) The line of lines is also of the same length, containing ten equal parts which are numbered, and which are again subdivided. The radius of the lines of secants, and of tangents above 45°, is about one fourth of the length of the other lines. From the end of the radius, which for the secants is at 0, and for the tangents at 45°, these lines extend to between 70° and 80°. The line of polygons is numbered 4, 5, 6, &c. from the extremity of each scale, towards the center.

The simple principle on which the utility of these several pairs of lines depends is this, that the sides of similar triangles are proportional. (Euc. 4. 6.) So that sines, tangents, &c. are furnished to any radius, within the extent of the opening of the two scales. Let AC and AC' (Fig. 40.) be any pair of lines on the sector, and AB and AB' equal portions of these lines. As AC and AC' are equal, the triangle ACC' is isosceles, and similar to ABB'. Therefore,

AB: AC::BB': CC'.

Distances measured from the center on either scale, as AB and AC, are called lateral distances. And the distances between corresponding points of the two scales, as BB' and CC', are called transverse distances.

Let AC and CC' be radii of two circles. Then if AB be the chord, sine, tangent, or secant, of any number of degrees in one; BB' will be the chord, sine, tangent, or secant, of the same number of degrees in the other. (Art. 119.) Thus, to find the chord of 30°, to a radius of four inches, open the sector so as to make the transverse distance from 60 to 60, on the lines of chords, four inches; and the distance from 30 to 30, on the same lines, will be the chord required. To find the sine of 28°, make the distance from 90 to 90, on the lines of sines, equal to radius; and the distance from 28 to 28 will be the sine. To find the tangent of 37°, make the distance from 45 to 45, on the lines of tangents, equal to radius; and the distance from 37 to 37 will be the tangent. In finding secants, the distance from 0 to 0 must be made radius. (Art. 201.)

To lay down an angle of 34°, describe a circle, of any convenient radius, open the sector, so that the distance from 60 to 60 on the lines of chords shall be equal to this radius, and to the circle apply a chord equal to the distance from 34 to 34. (Art. 161.) For an angle above 60°, the chord of half the number of degrees may be taken, and applied twice on the arc, as in art. 161.

The line of polygons contains the chords of arcs of a circle which is divided into equal portions. Thus the distances from the center of the sector to 4, 5, 6, and 7, are the chords of,,, and of a circle. The distance 6 is the radius. (Art. 95.) This line is used to make a regular polygon, or to inscribe one in a given circle. Thus, to make a pentagon with the transverse distance from 6 to 6 for radius, describe a circle, and the distance from 5 to 5 will be the length of one of the sides of a pentagon inscribed in that circle.

The line of lines is used to divide a line into equal or proportional parts, to find fourth proportionals, &c. Thus, to divide a line into 7 equal parts, make the length of the given line the transverse distance from 7 to 7, and the distance from 1 to 1 will be one of the parts. To find of a line, make the transverse distance from 5 to 5 equal to the given line; and the distance from 3 to 3 will be of it.

In working the proportions in trigonometry on the sector, the lengths of the sides of triangles are taken from the line

of lines, and the degrees and minutes from the lines of sines, tangents, or secants. Thus in art. 135, ex. 1,

35: R: 26: sin 48°.

To find the fourth term of this proportion by the sector, make the lateral distance 35 on the line of lines, a transverse distance from 90 to 90 on the lines of sines; then the lateral distance 26 on the line of lines, will be the transverse distance from 48 to 48 on the lines of sines.

For a more particular account of the construction and uses of the Sector, see Stone's edition of Bion on Mathematical Instruments, Hutton's Dictionary, and Robertson's Treatise on Mathematical Instruments.


NOTE H. p. 124.

The error in supposing that arcs less than 1 minute are proportional to their sines, cannot affect the first ten places of decimals. Let AB and AB' (Fig. 41.) each equal 1 minThe tangents of these arcs BT and B'T are equal, as are also the sines BS and B'S. The arc BAB' is greater than BS+B'S, but less than BT+B'T. Therefore BA is greater than BS, but less than BT: that is, the difference between the sine and the arc is less than the difference between the sine and the tangent.

Now the sine of 1 minute is


And the tangent of 1 minute is 0.000290888204

The difference is


The difference between the sine and the arc of 1 minute is less than this; and the error in supposing that the sines of 1', and of 0' 52" 44"" 3""" 45"""" are proportional to their arcs, as in art. 223, is still less.

NOTE I. p. 125.

There are various ways in which sines and cosines may be more expeditiously calculated, than by the method which ist

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