becomes the subject of investigation. In the one case, the various powers of the unknown quantity enter into the equation; in the other, the different orders of the function are concerned."1

When Harriott transposed the absolute term of an equation to the same side as the other terms which involve the unknown quantity, he was enabled to show that an equation of any degree could be formed with simple equations; and conversely, that every equation has as many roots as are denoted by the index of the highest power of the unknown quantity. From that time the problem for the solution of equations became the discovery of some general method by which the roots could be readily found. Solutions of certain forms of cubic and biquadratic equations had been devised by Tartaglia and Ferrari; but a like success has not attended the attempts of mathematicians to solve equations of the fifth and higher degrees.

In his researches in numerical equations Newton was led to seek the limits within which the real roots were situated, and then to determine their value by a method of approximation which bears his name. The difficulties and imperfections of this method were pointed out by La Grange in his "Traité de la Resolution des Equations Numériques de tous les degrés," which was first published in 1798; and he exhibited a method whereby the roots could be found in the form of continued fractions.

During the present century numerous attempts have been made to discover a general method of finding the roots of any equation, at the same time short and easy in its application. The valuable contributions of M. Budan were published in 1803, and those of M. Fourier, after his death in 1831, by M. Navier.

In this brief sketch it would not be possible to notice all the ingenious and curious methods which have been proposed to simplify or effect the solution of numerical equations. There are, however, two methods which appear to have superseded all preceding attempts

in this direction.

Mr. Horner's method of approximating to the roots of numerical equations first appeared in the Transactions of the Royal Society for the year 1819, and afterwards in other scientific publications. It consists of a general process, simple and easy, of approximation, for finding the roots of numerical equations. As in the extraction of the roots of numbers, so in this general evolution of the roots of equations; the first figure of a root is found by trial, and the other figures, one by one, are determined by successive steps. The first figure of the root sought, with its algebraical sign, is first required to be known before the method can be applied.

An important theorem of M. Sturm was printed in the sixth volume

1 1 Phil. Trans., 1815; Babbage, p. 339.

for 1835 of the "Mémoires présentés par divers Savans à l'Académie Royale des Sciences de l'Institut de France."

The mathematical prize for 1834 of the French Academy was awarded to M. Sturm on account of this discovery. His memoir exhibits a method of separating the real from the imaginary roots of numerical equations, the negative as well as the positive, and of finding the integral limits between which the roots are situated.

It will hence appear that, by a combination of the methods of Mr. Horner and M. Sturm, the number and situation of the real roots of any numerical equation can be ascertained, and approximate numerical results may be found to any degree of accuracy required.

Some of the recent extensions and additions to the science of Algebra, on account of their importance, require not to be passed over in silence. Dr. Salmon, at the end of the third edition of his work, entitled, “Lessons introductory to the Modern Higher Algebra," has added an historical account of the names and methods, and of the formations and properties of these functions. He states that the first idea of determinants, one class of these functions, originated with Leibnitz, and the method fell into oblivion until Cramer, in 1750, rediscovered the idea, and exhibited the determinants arising from linear equations. The subject since that time has engaged the attention of mathematicians, and considerable advancements have been made at different times. The more recent discoveries of Professor Cayley and Professor Sylvester have contributed largely to the completion and perfection of these new methods, as may be seen in their contributions which have appeared in the Philosophical Transactions and other scientific journals. Besides Dr. Salmon's "Lessons," other elementary treatises on this subject have been published, of which may be mentioned Spottiswoode's Elementary Theorems relating to Determinants, London, 1851; Brioschi on the same subject, Pavia, 1854; and Baltzer's work, Leipzig, 1857.

it

In closing these brief notices respecting the history of Algebra, may not be altogether useless to add a few remarks for the consideration of intelligent students who are desirous of securing the advantages to be gained from the study of the mathematical sciences. It has been remarked by a distinguished writer, "that mathematical studies, judiciously pursued, form one of the most effective means of developing and cultivating the reason."

There is a necessary connection between cause and effect, and that the same causes uniformly produce the same effects, and that the same effects will uniformly follow the action of the same causes in the future as have been observed in the past, are admitted as the fundamental principles of all reasonings, whether deductive or inductive.

All reasoning must rest upon facts or primary intuitive conceptions which are admitted from the evidence contained in themselves as obvious to the senses and the reason; and are simply admitted, and believed as true, because it is impossible to disbelieve them without manifest absurdity.

There are also distinctions between mathematical and moral evidence, as there are between conclusive and cumulative arguments; but the modes of reasoning, and the forms of argument, and the laws of inference, are the same in all subjects of human inquiry.

In the appropriate language of Dr. Whewell

"The chains of the logician generally consist only of two or three links. In the mathematics, on the contrary, every theorem is an example of an extended progressive chain; every proof consists of series of assertions, of which each depends on the preceding, but of which the last inferences are no less evident, or no less easily applied, than the simplest first principles. The language contains a constant succession of short and rapid references to what has been proved already, and it is justly assumed that each of these brief movements helps the reasoner forward in a course of infallible certainty. Each of these hasty glances must possess the clearness of intuitive evidence and the certainty of mature reflection, and yet must leave the reasoner's mind entirely free to turn instantly to the next step of his progress.

The late Professor Sedgwick, in a discourse he delivered in the College Chapel in 1831, on the occasion of the annual commemoration of the founder and benefactors of Trinity College, exhorted the junior members of the college in the following words with respect to their mathematical studies:

"A study of the laws of nature for many years has been, and I hope ever will be, held up to honour in this venerable seat of the discoveries of Newton. But in this, as in every other field of labour, no man can put aside the curse pronounced on him-that by the sweat of his brow he shall reap his harvest. Before he can reach that elevation from whence he may look down and comprehend the mysteries of the natural world, his way is steep and toilsome, and he must read the records of creation in a strange, and to many minds, a repulsive language, which, rejecting both the senses and the imagination, speaks only to the understanding. But when this language is once learnt, it becomes a mighty instrument of thought, teaching us to link together the phenomena of past and future times; and gives the mind a domination over many parts of the material world, by teaching it to comprehend the laws by which the actions of material things are governed. To follow in this track, first trodden by the immortal Newton-to study this language of pure unmixed truth, is to be regarded not only as your duty, but your high privilege. It is no servile task, no ungene

rous labour. The laws by which God has thought good to govern the universe are surely subjects of lofty contemplation; and the study of that symbolical language by which alone these laws can be fully deciphered, is well deserving of your noblest efforts."

Lord Brougham has pointed out, in the following expressive language, how a student may acquire some certain knowledge of oneof the systems of worlds in the universe-that in which the sun is the central orb around which revolve the earth and the other planets of the system :—

"The reader of the Principia, if he be a tolerably good mathematician, can follow the whole chain of demonstration by which the universality of gravitation is deduced from the fact, that it is a power acting inversely as the square of the distance from the centre of attraction. Satisfying himself of the laws which regulate the motion of the bodies in trajectories around given centres, he can convince himself of the sublime truths unfolded in that immortal work, and must yield his assent to this position, that the moon is deflected from the tangent of her orbit round the earth by the same force by which the satellites of Jupiter are deflected from the tangent of theirs, the very same force which makes a stone unsupported fall to the ground. The reader of the 'Mécanique Céleste,' if he be a still more learned mathematician, and versed in the modern improvements of the calculus which Newton discovered, can follow the chain of demonstration by which the wonderful provision made for the stability of the universe

1 Pp. 10, 11, "A Discourse on the Studies of the University of Cambridge," by Adam Sedgwick, M.A., F.R.S., Woodwardian Professor, and Fellow of Trinity College. The Fifth Edition, with additions, and a Preliminary Dissertation. Cambridge, 1850.

The occasion of the first publication of this discourse of Professor Sedgwick is not without some interest. The writer can never forget the electric effect of the discourse on his own mind, and on the minds of some other undergraduates who were present on that occasion. We were all so deeply impressed with the excellency of the counsels that the professor had delivered in his eloquent and expressive language, that we consulted together, and agreed to request the professor to print his discourse. A requisition was drawn up and signed by some of the undergraduates who heard the discourse, and it was placed in the hands of Mr. Whewell, one of the tutors of the college, to present to the professor. It appears that when the requisition was placed in Mr. Whewell's hands, the professor had left college for the Christmas vacation. Mr. Whewell, however, lost no time in communicating with the professor, as will appear from the following extract of a letter of the date, Trinity College, Dec. 23, 1831 :—“My dear Sedgwick,-When you had scribbled down the last sentence of your sermon after the bell had stopt, and had succeeded by a sort of miracle in reading your pothooks without spectacles, omitting, however, half the sentences, and a quarter of the syllables of those which remained, I dare say you thought you had done marvellously well, and had completed, or more properly had ended, your task. In this, however, you were mistaken, as I hope soon to make you acknowledge. The rising generation, who cannot err, inasmuch as they will dis

is deduced from the fact, that the direction of all the planetary motions is the same-the eccentricity of their orbits small, and the angle formed by the plane of their orbits with that of the ecliptic acute. Satisfying himself of the laws which regulate the mutual actions of these bodies, he can convince himself of a truth yet more sublime than Newton's discovery, though flowing from it, and must yield his assent to the marvellous position that all the irregularities occasioned in the system of the universe by the mutual attraction of its members are periodical, and subject to an eternal law which prevents them from ever exceeding a stated amount, and secures through all time the balanced structure of a universe composed of bodies whose mighty bulk and prodigious swiftness of motion mock the utmost efforts of the human imagination. All these truths are to the skilful mathematician as thoroughly known, and their evidence is as clear, as the simplest proposition of Arithmetic is to common understandings. But how few are those who thus know and comprehend them!"1

course most wise and true sentences when you and I are laid in the alluvial soil, declare that their intellectual culture requires that you should print and publish I will give you a list on the other side of the names [25] of the persons your sermon. who have joined in expressing this wish. I undertook very willingly to communicate this their desire to your reverence, inasmuch as I thought your sermon full of notions, as the Americans speak, which it will be very useful and beneficial to put in their heads; or rather to call them out, for a great number of those good thoughts are already ensconced in the excellent noddles of our youngsters, like flies in a bookcase in winter, and require only the sunshine of your seniorial countenance to call them into life and volatility. I do not know anything which will more tend to fix in their minds all the good they get here, than to have such feelings as you expressed, at the same time the gravest and the most animating which belong to our position, stamped, upon a solemn and official occasion, as the common property of them and us. And I also think it of consequence that, when they on their side proffer their sympathy in such reflections, we, on ours, that is, in the present case, your dignified self, should not be backward in meeting them, by giving to all parties the means of returning to and dwelling upon these reflections. Such is my thinking about this matter, and therefore I have undertaken to urge their request; and I hope you will be able to extract from some abyssmal recess your manuscript, and to place it before the astonished eyes of the compositor. It is probable that he will look, as Dante says the ghosts looked when they peered at him, like an old cobbler threading his needle,—but never mind that. The fronts of compositors were made to be corrugated by good sentences written in most vile hands; so let him fulfil his destiny without loss of time."

This extract is taken from Mr. Todhunter's account of the writings and selections from Dr. Whewell's correspondence, pp. 149, 150, vol. ii.

Professor Sedgwick printed his discourse and presented a copy of it to each of the undergraduates who signed the requisition.

1 Pp. 172, 173, vol. ii., "Dissertations on Subjects of Science connected with Natural Theology: being the concluding volumes of the new edition of Paley's Natural Theology." By Henry Lord Brougham, F.R.S., and Member of the National Institute of France. 2 vols. London. 1839.