114 



P O R I S M S. 



Porisms. 



In the same figure, the sums of the squares of the 

 perpendiculars are represented by 

 2 (a v) 2 + 2 (2 a v 3 w 2 -f- r 2 ) cos. <> 2 _|_ 4 u 2 cos. f* 



\- 



TC 

 4 V 8 COS. -I 



Q 2 



2 (a t>) 2 +2(2av 3 w 2 + r 2 ) cos. -J * 



4 v- cos. f -f 



2 2 



And the sums of the vertical series being found by 

 means of the formula already given, we have for the 

 sum of the squares 



2 n(a u) 2 + n (2 a u- 



which is independent of the value of 6; it may there- 

 fore 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 

 number 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 perpendiculars shall be equal to the square which is 

 * given. 



In this, as in many of the preceding propositions, 

 the given quantities must be contained within certain 

 limits, otherwise they may become impossible: these 

 limits will always be pointed out distinctly by the alge- 

 braic analysis. 



The theorems relating to the sums of series of the 

 powers of sines or cosines of arcs in arithmetical pro- 

 gression, 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 

 ellipse, or to any other combination of lines or curves,) 

 can be expressed in a rational integral form, with re- 

 spect 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 presumed, have demonstrated that this instrument of 

 investigation is at least as fertile in the number of the 

 conclusions to which it leads, as that analysis which 

 was contrived by 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 number might 

 easily be enlarged to an unlimited extent. Such, how- 

 ever, 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 are therefore more easily conceived, and their re- 

 lations more calculated to excite and fix the attention : 

 this, together with the imperfection of those instruc- 



tions which are usually given to the learner at the com- Porisms. 

 mencement of his algebraic studies, seems to be the real "^--y ' 

 ground of the asserted superiority of geometrical over 

 algebraical reasoning, for the purpose to which we have 

 alluded. 



PORSON, RICHARD, a celebrated Greek scholar, 

 was born at East Ruston in Norfolk, on the 25th 

 Dec. 1759. He was the son of Mr. Huggin Porson, 

 the parish clerk, who taught him arithmetic, reading, 

 and writing. He learned the alphabet by tracing the 

 letters upon sand, or upon a board; and he acquired 

 his arithmetic without the aid of a book or slate. 



In the ninth year of his age he was sent along with 

 his brother Thomas to the village school, kept by Mr. 

 Summers, under whom he continued three years. 

 The Rev. Mr. Hewitt, having heard of his uncommon 

 proficiency in arithmetic and other elementary studies, 

 undertook to instruct Richard in classical knowledge; 

 and Mr. Norris, a neighbouring country gentleman, 

 was at the expence of sending him to Eton in 1 772. 

 By the death of Mr. Norris, he was thrown upon the 

 liberality of some friends, particularly Sir George 

 Baker, who took him into his house during the vaca- 

 tion, and by receiving small subscriptions, purchased 

 an annuity of 80 per annum, for a few years, so as to 

 enable him to remain at Eton. .At this seminary he 

 was distinguished for his diligence and classical attain- 

 ments, but particularly by his extraordinary memory, 

 which enabled him to bring forward all that he had 

 read. The receipt of a copy of Toup's Longinus, 

 given him as a reward for a good exercise, is said 

 to have first given him a decided turn for critical en- 

 quiries. 



Towards the end of the year 1 777, he was sent to 

 Trinity College, where he obtained the classical prize 

 medal, and the university scholarship. In 1781 he 

 obtained a fellowship in Trinity College, and in 1785 

 he took his degree of master of arts. He declined, 

 however, to sign the Thirty-nine Articles; and being 

 thus unable to take orders, he was necessarily depriv- 

 ed of his fellowship in 1791. By means of a subsgrip- 

 tion, an annuity of 100 during his life was purchas. 

 ed for him, and the addition of 40 per annum was 

 made to his income, by his appointment to the profes- 

 sorship of Greek at Cambridge. He was married in 

 the year 1795, but losing his wife in 1797, his own 

 health was observed to decline. A spasmodic asthma, 

 increased by irregularity in his mode of life, inter- 

 rupted in a serious manner the laborious studies which 

 he had been in the habit of pursuing. 



Upon the establishment of the London institution, 

 he was appointed principal librarian, with a salary of 

 200 per annum; but he did not long enjoy this 

 comfortable sinecure. His former complaint renewed 

 its attacks, and weakened his frame; and in consequence 

 of an apoplectic stroke, he expired on the 25th Sep- 

 tember, 1808, in the forty-ninth year of his age. His 

 body was removed to Cambridge, where it was re- 

 ceived by the Bishop of Bristol with every mark of 

 respect, and interred in the chapel of Trinity College, 

 near the remains of Bentley. 



The principal writings of Porson are, his letters to 

 Mr. Archdeacon Travis, in answer to his Defence of 

 the Three Heavenly Witnesses ; his MSS. of Pho- 

 tius's Lexicon, which appeared at Cambridge in 1822, 

 and his four plays of Euripides, with the prefaces, 

 viz. Hecuba, which appeared at London in 1797; 

 Orestes, Lond. 1798, 1811; Medea, Cambridge, 1801, 

 Lond. 1812. The whole together appeared at Lon- 

 don in 1822. 



