From this, we deduce, as an immediate consequence, the proposition which forms the object of this Note. Let be the semicircumference of which the radius is 1; if were rational, the arc would be so too, and

therefore its tangent would be irrational: but the tangent of the arc is 4

well known to be equal to the radius 1; hence cannot be irrational. Hence the ratio of the circumference to the diameter is an irrational number.*

It is probable that this number is not even included among algebraical irrational quantities, in other words, that it cannot be the root of an algebraical equation having a finite number of terms with rational co-efficients: but a rigorous demonstration of this seems very difficult to find; we can only show that the square of is also an irrational number.

Thus, if in the continued fraction, which denotes tang. x, we put x=”, since tang. x=0, we must have

Now, as this continued fraction evidently comes under Lemma II., its value also must be irrational, and cannot be equal to the number 3. Hence the square of the ratio between the circumference and the diameter is an irrational number.

* This proposition was first demonstrated by Lambert, in the Memoirs of Berlin, anno 1761.

TREATISE

ON

TRIGONOMETRY.

TRIGONOMETRY has for its object the solution of triangles, that is, the determination of their sides and angles, when a sufficient number of those sides and angles is given.

In rectilineal triangles, it is sufficient to know three of the six parts which compose them, provided there be a side among these three. If the three angles only were given, it is obvious that all similar triangles would answer the question.

In spherical triangles, any three given parts, angles or sides, are always sufficient to determine the triangle; because, in triangles of this sort, the absolute magnitude of the sides is not considered, but only their relation to the quadrant, or the number of degrees which they contain.

In the Problems annexed to Book II., we have already seen how rectilineal triangles are constructed by means of three given parts. Propositions 361 and 363 of Book V. give likewise an idea of the constructions, by which the analogous cases of spherical triangles might be resolved. But those constructions, though perfectly correct in theory, would give' only a moderate approximation in practice, on account of the imperfection of the instruments required in constructing them they are called graphic methods. : Trigonometrical methods, on the contrary, being independent of all mechanical operations, give solutions with the utmost accuracy: they are founded upon the properties of lines called sines, cosines, tangents, &c. which furnish a very simple mode of expressing the relations that subsist between the sides and angles of triangles.

* We are naturally required to distinguish the figures which serve only to direct our reasoning in the demonstration of a theorem or the solution of a problem, from the figures which are constructed to find some of their dimensions. The first are always supposed to be exact; the second, if not exactly drawn, will give false results.

We shall first explain the properties of those lines, and the principal formulas derived from them; formulas which are of great use in all the branches of mathematics, and which even furnish means of improvement to algebraical analysis. We shall next apply those results to the solution of rectilineal triangles, and then to that of spherical triangles.

DIVISION OF THE CIRCUMFERENCE.

I. For the purposes of trigonometrical calculations, the circumference of the circle is conceived to be divided into 360 equal parts, called degrees; each degree into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds.

The semicircumference, or the measure of two right angles, contains 180 degrees; the quarter of the circumference, usually denominated the quadrant, and which measures the right angle, contains 90 degrees.

II. Degrees, minutes, and seconds, are respectively designated by the characters: thus the expression 16° 6' 15" represents an arc, or an angle, of 16 degrees, 6 minutes, and 15 seconds.

III. The complement of an angle, or of an arc, is what remains after taking that angle or that arc from 90°. Thus an angle of 25° 40′, has for its complement 64° 20′; an angle of 12o 4' 32", has for its complement 77° 55′ 28′′.

In general, A being any angle or any arc, 90°-A is the complement of that angle or arc. Whence it is evident that, if the angle or arc is greater than 90°, its complement will be negative. Thus the complement of 160° 34′ 10′′ is — 70° 34' 10". In this case, the complement, positively taken, would be the quantity requiring to be subtracted from the given angle or arc, that the remainder might be equal to 90°.

The two angles of a right-angled triangle, are, together, equal to a right-angle: they are, therefore, complements of each other.

IV. The supplement of an angle, or of an arc, is what remains after taking that angle or arc from 180°, the value of two right angles, or of a semi-circumference. Thus A being any angle or arc, 180°— A is its supplement.

In any triangle, an angle is the supplement of the sum of the two others, since the three together make 180o.

The angles of triangles rectilineal and spherical, and the sides of the latter, have their supplements always positive; for they are always less than 180°.

GENERAL IDEAS RELATING TO SINES, COSINES, TANGENTS, &c.

V. The sine of the arc S AM, or of the angle ACM, is the perpendicular MP let fall from one extremity of the arc, on the diameter which passes through the other extremity.

If at the extremity of the radius CA, the perpendicular AT is drawn to meet the production of the radius CM, the line AT, thus ter

minated, is called the tangent, and CT the secant of the arc AM, or of the angle ACM.

These three lines MP, AT, CT, are dependent upon the arc AM, and are always determined by it and the radius; they are thus designated: MP=sin AM, or sin ACM, AT=tang AM, or tang ACM, CT sec AM, or sec ACM.

VI. Having taken the arc AD equal to a quadrant, from the points M and D draw the lines MQ, DS perpendicular to the radius CD, the one terminated by that radius, the other terminated by the radius CM produced; the lines MQ, DS and CS, will, in like manner, be the sine, tangent, and secant of the arc MD, the complement of AM. For the sake of brevity, they are called the cosine, cotangent, and cosecant, of the arc AM, and are thus designated: MQ=cos AM, or cos ACM, DS=cot AM, or cot ACM, CS=cosec AM or cosec ACM. In general, A being any arc or angle, we have cos A =sin (90°-A), cot A=tang (90°-A), cosce A=sec (90°— A).

The triangle MQC is, by construction, equal to the triangle CPM; consequently CP=MQ: hence in the right-angled triangle CMP, whose hypotenuse is equal to the radius, the two sides MP, CP are the sine and cosine of the arc AM. As to the triangles CAT, CDS, they are similar to the equal triangles CPM, CQM; hence they are similar to each other. From these principles, we shall very soon deduce the different relations which exist between the lines now defined: be

fore doing so, however, we must examine the progressive march of those lines, when the arc to which they relate increases from zero to 180°.

VII. Suppose one extremity of the arc remains fixed in A, while the other extremity, marked M, runs successively throughout the whole extent of the semicircumference, from A to B in the direction ADB.

When the point M is at A, or when the arc AM is zero, the three points T, M, P, are confounded with the point A; whence it appears that the sine and tangent of an arc zero, are zero, and the cosine and secant of this same arc, are each equal to the radius. Hence if R represent the radius of the circle, we have

sin o=0, tang o=o, cos o=R, sec o=R.

VIII. As the point M advances towards D, the sine increases, and likewise the tangent and the secant; but the cosine, the cotangent, and the cosecant, diminish.

When the point M is at the middle of AD, or when the arc^ AM is 45°, and also its complement MD, the sine MP is equal to the cosine MQ or CP; and the triangle CMP, having become isosceles, gives the proportion MP: CM:: 1: √2, or

sin 45° R::1: √2. Hence sin 45°-cos 45°:

:

In this same case, the triangle CAT becomes isosceles and equal to the triangle CDS; whence, the tangent of 45° and its cotangent, are each equal to the radius, and consequently we have tang 45°=cot 45°=R.

IX. The arc AM continuing to increase, the sine increases till M arrives at D; at which point the sine is equal to the radius, and the cosine is zero. Hence we have sin 90°=R, cos 90°=0; and it may be observed, that these values are a consequence of the values already found for the sine and cosine of the arc zero; because, the complement of 90° being zero, we have sin 90°=cos o°=R, and cos 90°=sin o°=o.

As to the tangent, it increases very rapidly as the point M approaches D; and finally when this point reaches D, the tangent properly exists no longer, because the lines AT, CD, being parallel, cannot meet. This is expressed by saying that the tangent of 90° is infinite; and we write tang 90°= ∞ .

The complement of 90° being zero, we have tang o=cot, 90° and cot o=tang 90°. Hence cot 90°=o, and cot o=∞.