COS. + COS. 0 71 .cos n = 2 7t1 2 1 . 771 + I I. 2 771 22'.
where nz must be less than 71 ; if we apply these conside rations to the several series in the above sum, which re spectively multiply 2 -0 and we shall find for the value of the and for that which multiplies V ; so 8 that the sum of all the fourth powers of the chords is n 71 + 6 n If in is less than n, then it may be readily shown that if chords be drawn through any point within a circle, making equal angles with each other, the sum of the 2 in powers of those chords will always be equal to a constant quantity; for the 2 in powers of such chords are 22m sin. 9 sin. 0 2 71 2 ") , (ra sin.8 - 7/ And if these expressions arc expanded, the series of the powers of sines, which constitute each vertical column, are each equal to some constant quantity : the whole sum is therefore independent of 0.
A straight line and a circle being given, and also a point in that diameter of the circle which is perpendicular to the given line, it has been found that the sum of the perpen diculars to the given line drawn from the extremities of any chord passing through the given point, is expressed by 2 (a v + v cos. If another chord pass through the same point at right angles to the former, the expression for the sum of the perpendiculars drawn from its extremities to the given line, will be 2 (a v + -v sin. The sum of the four perpendiculars is therefore 4 (n + 2 v = 4 a 2 v.
This produces the following porism : A circle and a straight line being given, a point may be found within the circle, such that, if any two chords be drawn through it at right angles to each other, and if front the extremities of these chords perpendiculars be drawn to the given line, the sum of these four perpendiculars shall be equal to a given right line.
This property may be generalized, by supposing n chords, instead of two, to be drawn through the point found, and it will be perceived, that, if they make equal angles round that point, the sums of the perpendiculars drawn from their extremities to the given line will be a con stant quantity.
In the same figure, the sums of the rectangles under the perpendiculars, let fall from each chord, will be thus ex pressed : which is independent of the value of 6; it may therefore be enunciated thus : A circle and a right line being given in position, and also a square being given ; a point within the circle may be found, such that, if. through the point found a given num ber n of chords be drawn, making equal angles with each other, and from their extremities perpendiculars be drawn to the given line, the sum of the squares of all these per pendiculars shall be equal to the square which is given.
In this, as in many of the preceding prepositions, the given quantities must be contained within certain limits, otherwise they may become impossible : these limits will always be pointed out distinctly by the algebraic analysis.
The theorems relating to the sums of series of the powers of sines or cosines of arcs in arithmetical progres sion, which have been noticed in a former page, are of very extensive use in the discovery of porisms. In fact, whatever line (whether relating to the circle or ellipsc,or to any other combination of lines or curves,) can be expressed in a rational integral form, with respect to the powers of the sine or cosine, if only the angle be continually increased by some aliquot part of the circumference, until it returns into the first line then the sum of all these lines will be constant, as will also the sums of any powers of such lines.
The explanation which has been given of the method of applying algebraic reasoning to the discovery of the most elegant class of geometrical propositions, will, it is pre usmed, have demonstrated that this instrument of investi gation is at least as fertile in the number of the conclusions to which it leads, as that analysis which was contrived the celebrated restorer of the lost books of Euclid. In comparing the time and attention which must be expended in employing the geometrical method with that which is requisite for the complete success of the algebraic analysis, the superior value of the latter is strikingly pre-eminent ; and if, by adding to the number of these propositions, any considerable benefit would accrue to the science, that num ber might easily be enlarged to an unlimited extent. Such, however, is not the case; and it must be acknowledged that these truths, in their geometrical form, are useful only for the purpose of cultivating those mental habits which mathematical studies tend so strongly to promote. Signs and figure are less abstract than mere number, and arc therefore more easily conceived, and their relations more calculated to excite and fix the attention : this, together with the imperfection of those instructions which are usually given to the learner at the commencement of his algebraic studies, seems to be the real ground of the asserted superiority of geometrical over algebraical reasoning, for the purpose to which we have alluded.