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(28.) It appears from article 1, that whatever number of arcs may be described and intercepted between the lines containing any angle, (which are called its legs or sides,) they contain an equal number of divisions or degrees, &c. If the arc is large, the divisions are of course proportionally large, as the number of such divisions is the same in all circles; and it is found in fact, that the size of any division is in exact proportion to the radius of the
circle, or to the length of a string with which every circle may be conceived to be described. This truth is expressed technically, by saying, the circumferences of circles, and corresponding parts of them, vary as their radii.
(29.) Certain relations subsist between the sides and angles of all triangles; and they are determined through the medium of circles and right angled triangles. Construct fig. 5, by drawing the three circles having the same centre C, Fig. 5.
and their diameters at right angles to each other, as there represented; draw the line C M M' M", and from the points M, M', M" draw the lines M F, M' F', M" F" perpendiculars to the diameter D" A", then the three right angled triangles FM C, F' M' C, and FM" C will be equiangular, that is, will have all the angles of the one equal to all the angles of the other, each to each; for the right angle in every one is of course equal to the right angles in the others, and the angle at C forms the angle at the base to every one of the three triangles, that is, it is common to all the three; and as all the angles of a plane triangle are together equal to two right angles (Art. 5) the remaining or third angle must be equal in all the triangles; for that angle is the complement (Art. 5) of the angle at C in each of the triangles. Now all plane triangles which are equiangular, have the sides which contain the corresponding equal angles proportional; that is, the sides which contain the right angle in
From the Latin word radius, (the plural of which is radii,) which signifies the spoke of a wheel; these radii are of course all equal to one another.
each are proportional to the sides containing the right angles in the others, and the same is true of the other angles;* hence CM: MF:: CM': M'F, CM: MF:: CM": M" F", and CM': M' F' :: CM": M" F", and the same is true of lines similarly drawn in any number of circles; but the lines C M, Č M', and CM", are radii (see Art. 28) of the respective circles, and the lines MF, M'F', and M" F" are called the sinest of the angle at C; the line MF is called the sine of the angle at C to radius C M, M' F the sine of the same angle to radius CM', and M" F" the sine of the same angle to radius C M"; also the three equiangular triangles have the sides about the angle at C proportional, or CM: CF:: CM': CF:CM": CF";
Simson's Euclid, book 6. prop. 4.
triangles are equiangular, and have the sides containing corresponding equal angles proportional, the ing sides are called homologous, that is, having the triangles are said to be similar, and the correspondsame ratio. The proposition referred to in this note is one of the most useful and important in geometry.
+ The origin of this word is doubtful, it is pro-. bably from the Latin word sinus, which is itself a translation of the Arabian word Jeib a bosom, the Arabian name for a sine. See Preface to Hutton's Logarithms.
now the lines CF, CF, and CF" are termed the cosines of the angle at C to the three radii C M, C M', and CM"; so that the sine of any angle may be defined to be, a line drawn from one extremity of the arc on which the angle stands, perpendicular to a diameter of the circle to which the arc belongs, passing through the other extremity of the arc; and the cosine of an angle to be that part of the same diameter which is intercepted between the sine and the centre of the circle: the proportions before stated may now be more technically expressed as follows, radius (C M): the sine or cosine of any angle (M F or CF) to radius C M:: any other radius (CM): the sine or cosine of the same 'angle (M' F'or C F) to that other radius CM'. So that if we could only know the value of one sine or cosine of a certain number of angles, and the radius to which these sines or cosines were calculated, we should then have two terms in a proportion, or rule of three sum, and then, having a third, we should be able, without difficulty, to find the fourth term. Now, in fact, we do know the values of one radius and one sine and cosine to every second in the circle; we have tables calculated which give the proportional value of the sine and cosine of each angle to an assumed radius; and therefore when we wish to know the value of any other sine or cosine, the only other thing requisite to be known besides the angle itself, is the value of the radius, to which this latter sine or cosine is referred; and on the other hand, if we wish to know the value of a new radius, we must know the value of a sine or cosine to that radius. To illustrate this, suppose the angle at C = 43°, and the line C M or radius
in value to the radius in the tables, or unity; now as the tables give the value of the sine of any angle to the radius CM(1), if we wish to know the length of the line M'F, and we know the length of the line C M' (suppose CM'= 2) we can immediately find the line M'F' by the following proportion, CM (1); tabular sine of 43° (= 0,682) :: CM' (2): the answer = 1,364.
(30.) We are also enabled in the
same manner to ascertain the values
of certain other lines which we shall now proceed to define. The tangent of an angle or arc is a line drawn
The lines M F, C F, M' F', C F, &c. are called sines and cosines to the arcs, as well as to the angle which the arcs measure.
from that extremity of the arc through which the diameter of the circle passes, (which extremity is called the beginning of the arc,) perpendicular to that diameter, touching the circle but not cutting it,* and terminated by the radius of the circle produced, and passing through the other extremity of the are: the radius thus produced until it meets the tangent, is called the secant of the angle or arc, from the Latin word secans which signifies cutting; thus, in fig. 6, AT is the tangent of the arc A M
and angle AC M, CT the secant, MF the sine, and CF the cosine; also the difference between any arc and 90° is called the complement of that arc; and and 180°, is called the supplement of the difference between any arc or angle that are or angle; thus in fig. 6, a M and the angle a CM are the complements of the arc MA and the angle MCA, and DM and DCM are the Now it is easily demonstrated (see note supplements of the same arc and angle. Art. 14) that CF the cosine of the angle MCA
f M the sine of the angle a C M, and that M F the sine of the angle MCA=f C the cosine of the angle a CM; that is, the cosine of any arc or angle equals the sine of its complement, and the sine of any arc or angle and the lines C F, a t and Ct derive the equals the cosine of its complement ; names of cosine, cotangent, and cosecant, from the circumstance that they and secant, of the complement of the respectively represent the sine, tangent, arc or angle named; thus the line Ct is called the cosecant of MCA, or the secant of its complement MC a; and M CA, or the tangent of its complethe line at is the cotangent of the angle
From this circumstance the line derives its name, it being called tangent from the Latin word tangens, which signifies touching.
† Simson's Euclid, book 1. prop. 29 and 34,
(31.) Now, as the triangles FCM and ACT are similar, we have the proportion CF CM:: CA: CT, or cos.: rad. rad. sec.: and by similar triangles a Ct, MCF, Ct: a C:: CM: FM, or cosec. : rad. :: rad.: sin. Also (Art. 29 and 30) tabular rad.: tab. sin. cos. tan. or sec. :: new or given rad. : sin. cos. tan. or sec. to the new or given rad., or sin. cos. tan. or sec. to new or given tab. sin. &c. x new rad. rad. tab. rad. and if the tab. rad. = to the new rad. =
1, the sin. cos. &c. tab. sin. &c. x new rad. So that if any three parts of the right angled triangles C FM and CAT are given, a side being always one, the other three may be found.
(32.) There are two kinds of trigonometrical tables, the first kind contains the sines, cosines, &c. of angles, calculated to the radius unity. The sines, cosines, tangents, and secants in these tables to radius 1, are called natural, to distinguish them from those contained in the second description of trigonometrical tables, which are called artificial or logarithmic.
(33.) The artificial or logarithmic sines, &c. are far more frequently used than the natural; for, by a peculiar artifice, they are so constructed, that in all cases in which, were it not for this invention, it would be necessary to multiply natural sines, &c. by each other, we need only add the logarithmic sines, &c.; and, in cases
in which we should be compelled to divide a natural sine or cosine, &c. by another, we need only subtract the logarithmic sine, &c. which represents the divisor from the logarithmic sine, &c. A which represents the dividend; this contrivance, it will readily be seen, must often save much time and labour. There are also logarithms of numbers of the
The product of the second and third terms divided by the first. See the Treatise on Algebra, (Art. 127 and 128.)
From two Greek words, logos and arithmos, which signify the number of ratios or proportions, or the number showing the proportion.
same construction, and presenting the same facilities to the computer. This word logarithm in calculations is always abridged by writing log.; and the logarithmic sine of the arc A M is written, log. sin. A M.
We shall hereafter have occasion to employ these logarithms, and to illustrate their use and nature by examples.
The radius of the logarithmic tables is 10.000,000,000; our equation in Art. 31, when used with these tables, becomes log. new sin., &c. = log. tab. sin., &c. + log. new rad.-log. tab. rad. The logarithm of the radius in the artificial tables is 10.
(34.) There are also proportions, by means of which, when any three parts of an oblique angled plane triangle are given (except the three angles) the other three may be found. In the case of spherical triangles also, both right angled and oblique angled, there are proportions which enable us, three parts being given, to determine the other three: and with respect to spherical triangles there is this peculiarity, that even where the three angles only are given, the other parts, viz. the three sides, may be found, and the whole triangle determined.
(35.) A sphere is a solid shaped like a ball, or, to use the language of science, bounded by a curve surface of such a nature, that every point thereon is situated at an equal distance from a point within the solid called the centre. great circle of a sphere is that whose plane passes through the centre of the sphere; it divides the sphere into two equal parts, and has the same centre as the sphere itself. In fig. 7 the circle AEB CD, the plane of which passes Fig. 7.
through the centre F of the sphere APCP, is a great circle; PAP and PEP are respectively halves of great circles; all great circles of the same sphere are equal. The plane of a small circle, does not pass through the centre of the sphere. În fig. 7, abcd represents such a circle. A spherical triangle is conceived to be formed on the surface of some sphere, three great circles of which intersect each other in such a manner that their arcs enclose on that surface a triangular space; thus AEP is a spherical triangle. If a diameter of a sphere (PFp) be drawn per pendicular to the plane of a great circle of that sphere (A B C D), the two extremities of that diameter P and p are called the poles of that great circle. Every spherical angle is supposed to be contained between the arcs of great circles; and the angle thus formed on the surface of the sphere is equal to the angle contained between the planes of the great circles which form it; thus the spherical angle APE is equal to the plane angle contained between the semicircular planes PAP, PEp, at their intersection Pp; and it is measured by the arc A E of the great circle A E B CD intercepted between these two planes, the poles of which great circle are the two extremities P and p of that diameter, which forms the common intersection of the two planes.
A great circle passing through the poles of another great circle is called a secondary to it, and secondaries are perpendicular to the great circle to which they are secondaries; also the arcs of a secondary intercepted between the poles and the circumference of the great circle to which it is secondary, are all quadrants, or contain 90°. Thus the great circle PAPC, passing through the poles P and p of the great circle AEBCD is called a secondary to that circle; and the arcs PA and PC, pA and p C, as also the arcs of all other secondaries, intercepted between the poles P and p, and the circumference of the great circle A B C D, are quadrants.*
The above is to be considered merely as a short
view of the most practical and elementary part of trigonometry. The subject cannot be fully treated until the student has made considerable progress both in algebra and geometry. We refer him to the treatise on trigonometry, which will be published hereafter. In a subsequent part of the algebra the manner of constructing logarithmic tables, and their use, will be fully explained.
Definitions.-Latitude and Longitude -Magnitude of the Earth.
(36.) We shall take fig. 7 to represent the globe or sphere of the earth; * the small circle abcd is supposed to be parallel to the great circle A E B C D, and consequently perpendicular to its secondaries; a G c and AFC are diameters of those circles, and the point e is a point in which the small circle meets the great circle P E p.
(37.) This being premised, P and p may represent the north and south poles of the earth, the great circle A E B C D will then represent the equator, or equinoctial, or, as it is called by seamen, the line; the poles of which are the poles of the earth. All secondaries to the equator, as P Ap C, PEp, &c. are called meridians; the diameter PFp will represent the aris, round which the earth turns every 24 hours; and all small circles, as a b c d, parallel to the equator, are called parallels of latitude.
(38.) Let a, b, d, e, f, and g be six places on the earth's surface; then the meridian Pe Efp passing through the places e and f, is called the meridian of those places; and the meridian Pa Ap is called the meridian of the place a, and of all other places situated on the semicircle Pa Ap; for the meridian of any place is the secondary to the equator which passes through it; the arc of that secondary intercepted between the place and the equator is called the latitude of that place; thus the arc a A represents the latitude of the place a, the arcs e E and fE the latitudes of the places e and f. If the place is situated on the north side of the equator, its latitude is called north latitude; if on the south side, its latitude is called south latitude; thus e is in north latitude, and ƒ is in south latitude.
(39.) Places situated on the same parallel of latitude, as a and e, have the same latitude; for the arcs of secondaries, intercepted between their great circle and a small circle parallel to it, or between two small circles parallel to their great circle, are equal.
(40.) The longitude of any place is its distance from a particular meridian, called the first meridian, measured on
the arc of the equator, intercepted between the first meridian and the meridian of the place; suppose Pa Ap to be the particular meridian which is called the first meridian, then the arc of the equator A E will represent the longitude of the places e and f; for it follows from the above definition, that all places on the same meridian must have the same longitude.
(41.) If the meridian passing through the Royal Observatory at Greenwich, be that which is called the first meridian, places will be said to have east or west longitude, according as they are situated to the east or west of that meridian; thus, e, f, and b, have all east longitude, but a place d on the opposite side of the globe has west longitude.
(42.) The arc ef, intercepted between the two places e, f, represents their difference of latitude; and it follows from that which has preceded, that the are of any meridian or secondary to the equator, intercepted between the parallels of latitude passing through any two places, will represent their difference of latitude, (art 39;) thus, if gf be part of a parallel passing through fand g, ga will also represent the difference of latitude between e and f, and between g and e, and ƒ and a.
(43.) The difference of longitude between any two places is represented by the arc of the equator, intercepted between the meridians passing through the two places; which are also measures the spherical angle at the poles included between the two meridians, (Art. 35 ;) thus, the arc A E, or the spherical angle APE, represents the difference of longitude between the places a and e, and g and e, and ƒ and a.
(44.) The magnitude of the earth is that of a sphere of about 7916 English miles in diameter; and a sphere which has a diameter of that size, must be about 24869 English miles in circumference. Now the great circles of the earth are supposed to be divided into 360°, that is, into 21600', (Art. 1) and a geographical or nautical mile is of a degree of a great circle of the earth, or one minute of such a circle, (Art. 1 ;) therefore a great circle contains only 21600 nautical miles, whilst it contains 24869 common or statute miles; or a common mile is to a nautical mile, as 21600 to 24869; to express this proportion in feet we must say,
From the Greek word rave, naus, a ship.
: 5280 feet: 6079,089
Hence a geographical or nautical mile is
On the Mariner's Compass.-The half
(45.) THE course of a ship, or the angle