(Edward, M.D.), Lucasian Professor of Mathematics in the university of Cambridge, was the son of a wealthy farmer, of the Old Heath, near Shrewsbury. The early part of his education he received at the free school in Shrewsbury; whence he removed to Cambridge, and was admitted on the 24th of March 1753 a member of Magdalen college. Here his talents for abstruse calculation soon developed themselves, and at the time of taking his degree, he was considered as a prodigy in those sciences which make the subject of the bachelor's examination. The name of Senior Wrangler, or the first of the year, was thought scarcely a sufficient honour to distinguish one who so far outshone his contemporaries; and the merits of John Jebb were sufficiently acknowledged, by being the second in the list. Waring took his first, or bachelor's degree, in 1757, and the Lucasian Professorship became vacant before he was of sufficient standing for the next, or master's degree, which is a necessary qualification for that office. This defect was supplied by a royal mandate, through which he became master of arts in 1760; and shortly after his admission to this degree, the Lucasian Professor.

The royal mandate is too frequently a screen for indolence, and it is now become almost a custom, that heads of colleges, who ought to set the example in discipline to others, are the chief violators of it, by making their office a pretext for taking their doctor's degree in divinity, without performing those exercises which were designed as proofs of their qualifications. Such indolence cannot be imputed to Waring; yet several circumstances, previous to his election into the professorial chair, discovered that there was, at least, one person in the university who disapproved of the anticipation of degrees by external influence.—Waring, before his election, gave a small specimen of his abilities, as proof of his qualification for the office which he was then soliciting; and a controversy on his merits ensued: Dr Powell, the master of St John's college, attacking, in two pamphlets, the Professor; and his friend, afterwards Judge Wilson, defending. The attack was fearlessly warranted by the errors in the specimen; and the abundant proofs of talents in the exercise of the professorial office are the best answers to the sarcasms which the learned divine amused himself in casting on rising merit. An office held by a Barrow, a Newton, a Whiston, a Cotes, and a Vandermonde, must excite an ingenuous mind to the greatest exertions; and the new Professor, whatever may have been his success, did not fall behind any of his predecessors, in either zeal for the science, or application of the powers of his mind, to extend its boundaries. In 1762, he published his Miscellanea Analytica; one of the most abstruse books written on the abstrusest parts of algebra. This work extended his fame over all Europe. He was elected, without solicitation on his part, member of the societies of Bologna and Gottingen; and received flattering marks of esteem from the most eminent mathematicians at home and abroad. The difficulty of this work may be presumed from the writer's own words, "I cannot say that I know any one who thought it worth while to read through the whole, and perhaps not the half of it."

Mathematics did not, however, engross the whole of his attention. He could dedicate some time to the study of his future profession; and in 1767, he was admitted to the degree of doctor of physic; but, whether from the incapacity of uniting together the employments of active life with abstruse speculations, or from the natural diffidence of his temper, for which he was most peculiarly remarkable; the degree which gave him the right of exercising his talents in medicine was to him merely a barren title. Indeed he was so embarrassed in his manners before strangers, that he could not have made his way in a profession in which so much is done by address; and it was fortunate that the case of his circumstances permitted him to devote the whole of his time to his favourite pursuit. His life passed on, marked out by discoveries; chiefly in abstract science; and by the publication of them in the Philosophical Transactions, or in separate volumes, under his own inspection. He lived some years after taking his doctor's degree, at St Ives, in Huntingdonshire. While at Cambridge he married—quitted Cambridge with a view of living at Shrewsbury; but the air or smoke of the town being injurious to Mrs Waring's health, he removed to his own estate at Plaisley, about 8 miles from Shrewsbury, where he died in 1797, universally esteemed for inflexible integrity, modesty, plainness, and simplicity of manners. They who knew the greatness of his mind from his writings looked up to him with reverence everywhere; but he enjoyed himself in domestic circles with those chiefly among whom his pursuits could not be the object either of admiration or envy. The outward pomp which is affected frequently in the higher departments in academic life, was no gratification to one whose habits were of a very opposite nature; and he was too much occupied in science to attend to the intrigues of the university. There, in all questions of science, his word was the law; and at the annual examination of the candidates for the prize instituted by Dr Smith, he appeared to the greatest advantage. The candidates were generally three or four of the best proficient in the mathematics at the previous annual examination for the bachelor's degree, who were employed from nine o'clock in the morning to ten at night, with the exception of two hours for dinner, and twenty minutes for tea, in answering, viewing, or writing down answers to the professor's questions, from the first rudiments of philosophy to the deepest parts of his own and Sir Isaac Newton's works. Perhaps no part of Europe affords an instance of so severe a process; and there was never any ground for suspecting the Professor of partiality. The zeal and judgment with which he performed this part of his office cannot be obliterated from the memory of those who passed through his fiery ordeal.

Wishing to do ample justice to the talents and virtue of the Professor, we feel ourselves somewhat at a loss in speaking of the writings by which alone he will be known to posterity. He is the discoverer, according to his own account, of nearly 450 propositions in the analytics. This may appear a vain glorious boast, especially as the greater part of those discoveries are likely to sink into oblivion; but he was, in a manner, compelled to make it by the influence of Lalande, who, in his life of Condorcet, affirms that, in 1764, there was no first-rate analyst in England. In reply to this assertion, the Professor, in a letter to Dr. Markelyne, first mentions, with proper respect, the inventions and writings of Harriot, Briggs, Napier, Wallis, Halley, Brucker, Wren, Pell, Barrow, Mercator, Newton, De Moivre, Maclaurin, Cotes, Stirling, Taylor, Simpson, Emerson, Landen, and others; of whom Emerson and Landen were living in 1764. He then gives a fair and full detail of his own inventions, of which many were published anterior to 1764; and concludes his letter in these words:

"I know that Mr. Lalande is a first-rate astronomer, and writer of astronomy; but I never heard that he was much conversant in the deeper parts of mathematics; for which reason I take the liberty to ask him the following questions:

"Has he ever read or understood the writings of the English mathematicians; and, as the question comes from me, I subjoin, particularly of mine? If the answer be in the negative, as it is my opinion, if his answer be the truth, that it will, then there is an end of all further controversy—but if he affirms that he has, which is more than Condorcet did by his own acknowledgment, then he may know, from the enumeration of inventions made in the prefaces, with some subsequent ones added, that they are said to amount to more than 400 of one kind or other. Let him try to reduce those to as low a number as he can, with the least appearance of cadaver and truth; and then let him compare the number with the number of inventions of any French mathematician or mathematicians, either in the present or past times, and there will result a comparison (if I mistake not) not much to his liking; and, further, let him compare some of the first inventions of the French mathematicians with some of the first contained in my works, both as to utility, generality, novelty, difficulty, and elegance, but wisely as to utility, there is little contained in the deep parts of any science; he will find their difficulty and novelty from his difficulty of understanding them, and his never having read anything similar before; their generality, by the application of them; principles of elegance will differ in different persons.—I must say, that he will probably not find the difference expected. After or before this inquiry is instituted for mine, let him perform the same for the other English mathematicians; and when he has completed such inquiries, and not before, he will become a judge of the justice of his assertion; but I am afraid that he is not a sufficient adept in these studies to institute such inquiries; and if he was, such inquiries are invincible, troublesome, and of small utility."

By mathematical readers this account, which was not published by the Professor himself, is allowed to be very little, if at all, exaggerated. Yet if, according to his own confession, "few thought it worth their while to read even half of his works," there must be some grounds for this neglect, either from the difficulty of the subject, the unimportance of the discoveries, or a defect in the communication of them to the public. The subjects are certainly of a difficult nature, the calculations are abstruse; yet Europe contained many persons not to be deterred by the most intricate theorems. Shall we say then, that the discoveries were unimportant? If this were really the case, the want of utility would be a very small disparagement among those who cultivate science with a view chiefly to entertainment and the exercise of their rational powers. We are compelled, then, to attribute much of this neglect to a perplexity in style, manner, and language; the reader is stopped at every instant, first to make out the writer's meaning, then to fill up the chain in the demonstration. He must invent anew every invention; for, after the enunciation of the theorem or problem, and the mention of a few steps, little assistance is derived from the Professor's powers of explanation. Indeed, an anonymous writer, certainly of very considerable abilities, has aptly compared the works of Waring to the heavy appendages of a Gothic building, which add little of either beauty or stability to the structure.

A great part of the discoveries relate to an assumption in algebra, that equations may be generated by multiplying together others of inferior dimensions. The roots of these latter equations are frequently terms called negative or impossible; and the relation of these terms to the coefficients of the principal equation is a great object of inquiry. In this art the professor was very successful, though little assistance is to be derived from his writings in looking for the real roots. We shall not, perhaps, be deemed to depreciate his merits, if we place the series for the sum of the powers of the roots of any equation among the most ingenious of his discoveries; yet we cannot add, that it has very usefully enlarged the bounds of science, or that the algebraist will ever find occasion to introduce it into practice. We may say the same on many ingenious transformations of equations, on the discovery of impossible roots, and similar exertions of undoubtedly great talents. They have carried the assumption to its utmost limits; and the difficulty attending the speculation has rendered persons more anxious to ascertain its real utility; yet they who reject it may occasionally receive useful hints from the Miscellanea Analytica.

The first time of Waring's appearing in public as an author was, we believe, in the latter end of the year 1759, when he published the first chapter of the Miscellanea Analytica, as a specimen of his qualifications for the professorship; and this chapter he defended, in a reply to a pamphlet, intitled, "Observations on the First Chapter of a book called Miscellanea Analytica." Here the Professor was strangely puzzled with the common paradox, that nothing divided by nothing may be equal to various finite quantities, and has recourse to unquestionable authorities in proof of this position. The names of Maclaurin, Sanderson, De Moivre, Bernoulli, Monmort, are ranged in favour of his opinion: But Dr. Powell was not so easily convinced, and returns to the charge in defence of the Observations; to which the Professor Professor replied in a letter to the Rev. Dr Powell, fellow of St John's college, Cambridge, in answer to his Observations, &c. In this controversy, it is certain that the Professor gave evident proofs of his abilities; though it is equally certain that he followed too implicitly the decisions of his predecessors. No apparent advantage, no authority whatever, should induce mathematicians to swerve from the principles of right reasoning, on which their science is supposed to be peculiarly founded. According to Maclaurin, Dr Waring, and others, If \( P = \frac{a-x}{a^2-x^2} \), then, when \( x = a \), \( P \) is equal to \( \frac{1}{2a} \); for, say they, \( \frac{a-x}{a^2-x^2} \) is equal to \( \frac{a-x}{a-x} \times \frac{1}{a+x} \); that is, when \( x \) is equal to \( a \), \( P = \frac{1}{a+x} \), or \( \frac{1}{2a} \). But when \( x \) is equal to \( a \), the numerator and denominator of the fraction \( \frac{a-x}{a^2-x^2} \) are both, in their language, equal to nothing. Therefore, nothing divided by nothing is equal to \( \frac{1}{2a} \). In the same manner, \( \frac{a-x}{a^2-x^2} = \frac{1}{a^2+ax+x^2} \times \frac{a-x}{a-x} \), which, when \( x \) is equal to \( a \), becomes \( \frac{1}{3a^2} \). Therefore, nothing divided by nothing is equal to \( \frac{1}{3a^2} \), or \( \frac{1}{3a^2} = \frac{1}{2a} \); that is, \( \frac{1}{3a^2} = \frac{1}{2a} \); which is absurd. But we need only trace back our steps to see the fallacy in this mode of reasoning.

For \( P \) is equal to some number multiplied into \( \frac{a-x}{a-x} \); that is, when \( x \) is equal to \( a \), \( P \) is equal to some number multiplied into nothing, and divided by nothing; that is, \( P \) is, in that case, no number at all. For \( a-a \) cannot be divided by \( a-x \) when \( x \) is equal to \( a \), since, in that case, \( a-x \) is no number at all.

If, in the beginning of his career, the Professor could admit such paradoxes into his speculations, and the writings of the mathematicians, for nearly a century before him, may plead in his excuse, we are not to be surprised that his discoveries should be built rather on the assumptions of others than on any new principles of his own. Acquiescing in the strange notion, that nothing could be divided by nothing, and produce a variety of numbers, he has easily adopted the position, that an equation has as many roots as it has dimensions. Thus 2 and -4 are said to be roots of the equation \( x^2 - 2x = 8 \), though 4 can be the root only of the equation; \( x^2 - 2x = 8 \), which differs so materially from the preceding, that in one case \( 2x \) is added, in the other case it is subtracted from \( x^2 \).

Allowances being made for this error in the principles, the deductions are, in general, legitimately made; and any one, who can give himself the trouble of demonstrating the propositions, may find sufficient employment in the Professor's analytics. Perhaps it will be sufficient for a student to devote his time to the simplest case \( x^n + 1 = 0 \); and when he has found a few thousand roots of \( +1 \) and \( -1 \), the publication of them may afford to posterity a strong proof of the ingenuity of their predecessors, and the application of the powers of their mind to useful and important truths.

In this exercise may be consulted the method given by the Professor, of finding a quantity, which, multiplied into a given irrational quantity, will produce a rational product, or consequently exterminate irrational quantities out of a given equation; but if an irrational quantity cannot come into an equation, the utility of this invention will not be admitted without hesitation.

The "Proprietates Algebraicarum Curvarum," published in 1772, necessarily labour under the same defects with the Miscellanea Analytica, the Meditationes Algebraicae, published in 1770, and the Meditationes Analyticae, which were in the press during the years 1773, 1774, 1775, 1776. These were the chief and the most laborious works edited by the Professor; and in the Philosophical Transactions is to be found a variety of papers, which alone would be sufficient to place him in the first rank in the mathematical world. The nature of them may be seen from the following catalogue.

Vol. LIII. p. 294, Mathematical Problems.—LIV. 193. New Properties in Conics.—LV. 143. Two Theorems in Mathematics.—LXIX. Problems concerning Interpolations.—86. A General Refutation of Algebraical Equations.—LXXVI. 81. On Infinite Series. LXXVII. 71. On Finding the Values of Algebraical Quantities by Converging Series, and Demonstrating and Extending Propositions given by Pappus and others.—LXXVIII. 67. On Centripetal Forces.—LXXX. 588. On some Properties of the Sum of the Division of Numbers.—LXXIX. 166. On the Method of Correspondent Values, &c.—LXXXI. 185. On the Refutation of Attractive Powers.—LXXXI. 146. On Infinite Series.—LXXXIV. 385—415. On the Summation of those Series whose general term is a determinate function of \( z \), the distance of the term of the Series.

For these papers, the Professor was, in 1784, deservedly honoured by the Royal Society with Sir Godfrey Copley's medal; and most of them afford very strong proofs of the powers of his mind, both in abstract science, and the application of it to philosophy; though they labour, in common with his other works, under the disadvantage of being clothed in a very unattractive form. The mathematician, who has resolution to go through them, will not only add much to his own knowledge, but be usefully employed in dilating on those articles for the benefit of the more general reader. We might add in this place, a work written on morals and metaphysics in the English language; but as a few copies only were presented to his friends, and it was the Professor's wish that they should not have a more extensive circulation, we shall not here enlarge upon its contents.

In the mathematical world, the life of Waring may be considered as a distinguished era. The strictness of demonstration required by the ancients had gradually fallen into disuse, and a more commodious, though almost mechanical mode by algebra and fluxions took its place, and was carried to the utmost limit by the Professor. Hence many new demonstrations may be attributed to him, but so discoveries can scarcely fill to the lot of a human being. If we examine thoroughly those which our Professor would distinguish by such names, we shall find many to be mere deductions, o thers, as in the solution of biquadratics, anticipated by former writers. But if we cannot allow to him the merit of so inventive a genius, we must applaud his affability; and, distinguished as he was in the scientific world, the purity of his life, the simplicity of his manners, and the zeal which he always manifested for the truths of the Gospel, will entitle him to the respect of all who do not esteem the good qualities of the heart inferior to those of the head.