and height. Thus the area of the triangle ABG (Fig. 7) is stration may often be rendered much more simple and concise equal to half A B into G H, or its equal BC; that is,

H

B

Fig. 7.

A

E

G

L

B

Fig. 8.

D

than in ordinary language. The proposition (Euclid II. 4), that
when a straight line is divided into two parts, the square of the
whole line is equal to the squares of the two parts, together
with twice the product of the parts, is demonstrated by
squaring a binomial.

Let the side of a square be represented by s;
And let it be divided into two parts, a and b.
By the supposition,
s = a+b;
And squaring both sides, s2 = a2 + 2ab + b2.

That is, s2, the square of the whole line, is equal to a2 and b2,
the squares of the two parts, together with 2ab, twice the pro-
duct of the parts.

Fig. 9.

C

Algebraical notation may also be applied with great advantage to the solution of geometrical problems. In doing this it will be necessary, in the first place, to form an algebraical equation from the geometrical relations of the quantities given and required; and then by the usual reductions, to find the value of the

Thus the right-lined figure A B C D E (Fig. 8) is composed of unknown quantity in this equathe triangles A B C, A C E, and ECD.

The area of the whole figure is therefore equal to

(AC × BL) + (ACX EH) + ( ECX DG).

The expression for the superficies has here been derived from that of a line or lines. It is frequently necessary to reverse this order; to find a side of a figure, from knowing its area.

If the number of square inches in the parallelogram A B C D (Fig. 3), whose breadth, B C, is 3 inches, be divided by 3, the quotient will be a parallelogram, A B E F, one inch wide, and of the same length with the larger one. But the length of the small parallelogram is the length of its side, A B. The number of square inches in one is the same as the number of linear inches in the other. If, therefore, the area of the large parallelogram be represented by a, the side A B; that is, the length of a parallelogram is found by dividing the area by the breadth; and B C =

α AB

a BC

1. Given b the base, and a the sum of the hypothenuse and perpendicular of the right-angled triangle A B C (Fig. 9), to find the perpendicular B C.

Let the perpendicular BC. The sum of hypothenuse and perpendicular, a + A c = a, Then transposing a, A Ca— 2, (1.) By Euclid I. 47, (B C)2 + (A B)2 = (A C)2.

(2.) That is, by the notation, 2+ b2 = (a− x)2=a2— 2ax+s2.
a2- b2
And x =
BC, the side required. Hence,
2a

In a right-angled triangle, the perpendicular is equal to the square of the sum of the hypothenuse and perpendicular, dimin ished by the square of the base, and divided by twice the sum of the hypothenuse and perpendicular.

It is applied to particular cases by substituting numbers for the letters a and b. Thus, if the base is 8 feet, and the sum of the hypothenuse and perpendicular 16, the expression a2 b2 162-82 becomes = 6, the perpendicular; and this sub2a 2 x 16 tracted from 16, the sum of the hypothenuse and perpendicular, leaves 10, the length of the hypothenuse.

2. Given the base of a right-angled triangle A B C (Fig. 10) =b, and the difference between the hypothenuse and perpen. d, to find the perpendicular BC. Apply this where b 20 and d

That is, the side of the square is found by extracting the square dicular root of the number of measuring units in its area.

If A B be the base of a triangle, and BC its perpendicular height,

=

That is, the base of a triangle is found by dividing the area by half the height, and the height by dividing the area by half the base.

As a surface is expressed by the product of its length and breadth, the contents of a solid may be expressed by the product of its length, breadth, and depth. It is necessary to bear in mind, that the measuring unit of solids is a cube; and that the side of a cubic inch is a square inch; the side of a cubic foot, a square foot, etc.

Let ABCD (Fig. 3) represent the base of a parallelopiped, five inches long, three inches broad, and one inch deep. It is evident there must be as many cubic inches in the solid, as there are square inches in its base. And as the product of the lines A B and B C gives the area of this base, it gives, of course, the contents of the solid. But suppose that the depth of the parallelopiped, instead of being one inch, is four inches, its contents must be four times as great. If, then, the length be A B, the breadth BC, and the depth co, the expression for the solid contents will be ABX BCX CO.

By means of algebraical notation, a geometrical demon

poverty, privation, and embarrassment.

His education he received first at a grammar school at Kilkenny, and subsequently at Trinity College, Dublin. Here he not only failed to distinguish himself by his diligence or attainments, but seems to have left a very unfavourable impression of his abilities. Indeed, Swift's genius was very slow in showing itself: he was as remarkable an example of late mental development as his friend and fellow-worker, Pope, was of intellectual precocity. Swift was distantly connected by family with Sir William Temple; and not long after taking his degree, he entered the service of that statesman, then living in luxurious and lettered ease at his country seat in Surrey. Swift's employment in Temple's service was an ambiguous one, something between secretary, literary assistant, and humble hanger-on; and it may easily be conceived how acutely painful such a position must have been to Swift's proud, sensitive, and not very generous nature. There was everything, in fact, in Swift's early life and training to embitter such a disposition as his. And the facts of his history go far to explain how one capable of the depth of tenderness and affection which Swift could show, could yet have entertained that hatred and contempt for mankind which render his satire not severe merely, but positively savage and ferocious.

It was while in Temple's service that Swift first met Esther Johnson-then a very young girl, passing as the daughter of Temple's steward, though probably, in reality, a natural daughter of the old man himself. She was the Stella whose name must always remain associated with Swift's, and whose sad story is one of the most touching in the whole history of literature. An attachment seems early to have sprung up between Swift and her: on her side it ripened into an absolute and life-long devotion; on his side there was, as his Journal to Stella shows, an affection, a tenderness of the rarest kind; though with that strange, unaccountable cruelty, which was a part of his nature, he broke her heart through doubt, delay, and uncertainty, and married her only on her deathbed.

After the death of Sir William Temple, in 1699, it fell to the lot of Swift to collect and edit the works of his patron; and this appears to have been Swift's first public appearance in the paths of literature. He soon afterwards went to Ireland in the capacity, in the first instance, of chaplain to the then Lord Deputy, and was in time appointed to the living of Laracor in the county of Meath. This was now his home for some years; but his visits to London were frequent, where his great powers gradually became known, and his society proportionately cultivated among the wits and literary men of the metropolis.

His connection with Temple had naturally introduced him into political life as a Whig; but Swift's political principles were probably never very rigid, and before very long he took service under the Tory banner, and at once became the most powerful literary champion of the party of Harley and Bolingbroke.

It was during these constant visits to London that Swift's touching Journal to Stella was written, she remaining at that time near his home in Ireland. It was also during one of these visits that he became acquainted with the second victim of his affections, Esther Vanhomrigh, the daughter of a wealthy London merchant, who, under the poetical name of Vanessa given her by Swift, has become scarcely less famous than the unhappy Stella. Being left, by her father's death, with a competent independence, she also followed Swift to Ireland. Driven at last to desperation by doubt and jealousy, she sought to learn the truth about her rival, Stella (who was then, in truth, in her last illness, and whom Swift about the same time married), with a directness which excited his anger, and alienated him from her for ever. She died soon after, evidently under the influence of disappointed and wounded affection. In 1713 Swift had been appointed. to the Deanery of St. Patrick's Cathedral, Dublin; the character of his writings, and the personal enmity which his satire had in some instances excited, being an obstacle to that higher promotion to an English bishopric, which he so ardently desired and so confilently expected. During his residence in Dublin as dean, Swift showed his great powers as a satirist and party-leader in their most conspicuous light, and became almost in a moment the idol of the Irish nation. It had been determined by the Government to introduce a large quantity of a new copper coinage into Ireland; and an English manufacturer, named

Wood, had obtained the contract for the production of the new coin. Wood's halfpence were from the first regarded as a wrong and a fraud. But Swift took up the quarrel, and wrote his famous series of letters known as "Drapier's Letters," from their having been published under the signature of "M. B., Drapier." The skill with which these letters were framed was consummate, and their effect extraordinary. The people of Dublin, indeed of all Ireland, were excited to frenzy; the coinage had to be withdrawn; and though Swift was well known to be the author of the letters, the Government did not dare to attack him, and proceedings which had been commenced against the printer were discreetly abandoned. Thus did Swift "his wronged country's copper chains unbind."

But Swift's heart was never in Ireland. He was never an Irishman in real sympathy, and never loved to be thought one in any sense at all. London was the place to which his thoughts and wishes really turned; there he reigned supreme. He was courted by all the leading political men on both sides, and might have sold his services to either almost at his own price. In society his bitter and brilliant speech, and the dread of his powerful and somewhat unscrupulous pen, secured him that power which probably he valued more than affection. In the literary world he could have no rivals, except Pope and Addison. And Addison and Swift, though on opposite sides in politics, always treated one another at least with respect, a respect which Swift showed for few; and with Pope Swift lived on terms of close intimacy and genuine friendship.

Swift probably not only suffered throughout much of his life, but had even been conscious of a tendency to mental disorder; a tendency which may very probably be the true key to much of what is most strange and most painful in his very painful career. He had foretold in bitterness of spirit that he would die at top first." And so it was. Disease of the brain began to show itself in him in about 1741; and for the last four years of his life he was reduced to a state of absolute idiotcy, in which he died in 1745. He was buried in St. Patrick's Cathedral. By a strange freak of feeling, showing alike what the end he anticipated was, and how oddly that anticipation worked upon his mind, he left the bulk of his fortune to found an asylum for the insane in the city of Dublin, which still exists there under the name of Swift's Hospital.

To examine Swift's works with anything like the completeness which they deserve, would demand far more space than we can possibly give to them in these lessons. His poems are numerous, chiefly mero jeux d'esprit-occasional verses on the most trivial subjects. It is impossible that such a man as Swift can write anything that shall not have merit of a certain kind; but these are rather the works of a wit than of a poet.

Upon political and party questions Swift was a most powerful and not very scrupulous pamphleteer; though it must be admitted, that after he had once chosen the Tory side he remained faithful to that party. The most important of his controversial writings of this class is the celebrated pamphlet on "The Conduct of the Allies," published in 1712, a work which contributed largely to the fall of the Whig party, the abandonment of the Whig policy, and the triumph of Harley and Bolingbroke.

Others, again, of Swift's works seem to be almost purposeless, to be written in the very wantonness of satire, merely because it was a pleasure to "laugh and shake in Rabelais' easy chair," because he loved to show us the world turned upside down, to startle us with paradox, to shock our sensibilities, to bring all that is most venerable into contact with the most contemptible associations. Of this class are his "Argument against Abolishing Christianity," his "Modest Proposal to the Public," and his "Irrections to Servants."

But there are three in particular of Swift's works upon which his fame with posterity mainly rests: "The Battle of the Books," "The Tale of a Tub," both published in 1704; and "Gulliver's Travels," published in 1726.

The "Battle of the Books" is one of the many valuable pieces which we owe to the great discussion then at its height-of which the celebrated Boyle and Bentley controversy was an episode as to the relative merits of the ancients and the moderns in the field of literature. Sir William Temple had entered the arena as a champion of the ancients, and Swift, as became his humble dependent, was bound to take the same side. His work, regarded as a serious contribution to the literature

of the subject, would be of small value; it is neither learned nor critical. But as a squib, as a mere piece of abuse and ridicule of antagonists, it is in Swift's best style.

The Tale of a Tub" is one of the most extraordinary satires ever written. Its object is to ridicule extremes in religion, and exalt what in Swift's view was the happy medium of the High Church Anglican party. But few can, we think, read the "Tale of a Tub" without feeling that from the audacious levity with which the whole subject is handled, the coarse ridicule which is thrown over everything, the effect of this great work is not less hostile to religion itself than to the follies or eccentricities of any particular sect. The book tells us the adventures of three brothers, Peter, Martin, and Jack-representing the Roman Catholic, the Lutheran or moderate Protestant, and the Presbyterian bodies-left by their father with his written will to guide them, and professing, each of them, to govern their conduct by that will in every particular. That will stands for the New Testament; and the manner in which, in ordering his coat (his system of doctrine and practice) to suit his own taste and temper, each manages to find in the words or letters, or in the omissions of that will, authority for every ornament that he adds to, or every rent that he makes in the coat, is inexpressibly ludicrous. The book, too, is full of digressions, which show Swift's quaint, grotesque humour, and his infinite ingenuity of conception, in the strongest light. The "Tale of a Tub" is a masterpiece; but it is not difficult to understand that it may have stood, as it is said to have done, in the way of its author's promotion to a bishopric.

fresh opportunities for satire, the principal method which he adopts in this part being to show the ordinary affairs of life, such as Gulliver relates to the Giant King, in a ludicrous light, by placing them in contrast with another system social and political, incomparably grander in scale, and far simpler and purer. Thus, after Gulliver has with great pains and no little pride given the king a minute account of the state of England, we read that, "His Majesty in another audience was at the pains to recapitulate the sum of all I had spoken; compared the questions he made with the answers I had given; then taking me into his hands, and stroking me gently, delivered himself in these words, which I shall never forget, nor the manner he spoke them in: My little friend, Grildrig, you have made a most admirable panegyric upon your country; you have clearly proved that ignorance, idleness, and vice are the proper ingredients for qualifying a legislator; that laws are best er. plained, interpreted, and applied by those whose interest and abilities lie in perverting, confounding, and eluding them. I observe among you some lines of an institution, which in its original might have been tolerable, but these half erased, and the rest wholly blurred. and blotted by corruption. It does not appear, from all you have said, how any one perfection is required towards the procurement of any one station among you; much less that men are ennobled on account of their virtue; that priests are advanced for their piety or learning; soldiers, for their conduct or valour; judges, for their integrity; senators, for the love of their country; or councillors, for their wisdom.

As for yourself,' continued the king, who have spent the greatest part of your life in travelling, I am well disposed to hope you may hitherto have escaped many vices of your country. But by what I have gathered from your own relation, and the answers I have with much pains wringed and extorted from you, I cannot but conclude the bulk of your natives to be the most pernicious race of little odious vermin that Nature ever suffered to crawl upon the surface of the earth.'" In another place, the same king having heard from Gulliver a full history of the politics and state-craft of Europe, and the many books that have been written on the art of government, is filled with astonishment. "He professed both to abominate and despise all mystery, refinement, and intrigue, either in a prince or a minister. He could not tell what I meant by secrets of state, where an enemy or some rival nation were not in the case. He confined the knowledge of governing within very narrow bounds to common sense and reason, to justice and lenity, to the speedy determination of civil and criminal causes; with some other obvious topics which are not worth considering. And he gave it for his opinion, that whoever could make two ears of corn, or two blades of grass, to grow upon a spot of ground where only one grew before, would deserve better of mankind, and do more essential service to his country, than the whole race of politicians put together."

The most popular, however, and deservedly so, of Swift's works is the "Travels of Gulliver." It is one of the most comprehensive of satires. Swift, though one of the most original of thinkers, never hesitated to borrow from his predecessors, to several of whom he is largely indebted. But his chief master in satire was Rabelais, from whom he has derived not only much of his manner and style, but even many of his minutest details. "Gulliver," however, is wider on the whole in its scope than the great romance of Rabelais; it is less a satire upon particular classes, and more a satire upon human nature. The form which Swift chooses for his satire is one which had been adopted by others before, and has been since that of imaginary travels through strange regions. Lemuel Gulliver, a doctor by trade, and a traveller by taste of whose previous life and circumstances we are told just enough to give naturalness to the whole account-is shipwrecked, and escapes with bare life on an unknown shore, which turns out to be the kingdom of Lilliput, inhabited by a pigmy race not above six inches high. The description of Gulliver's adventures in Lilliput forms the first part of the work. With that peculiar power which Swift possessed of rendering every scene life-like by means of minute accuracy of detail, making everything which he sees in Lilliput relatively correct in size, he presents us with the most vivid picture of the world, with its kings and ministers, its courts, its politics, wars and intrigues, its pomp and splendour, all in miniature, and so all exposed in their utmost absurdity. Nothing can be more ludicrous, and at the same time more effective as satire, than the hereditary war between the Bigendians and the Smallendians-those who broke their eggs at the large end, and those who broke them at the small end; the two parties in the state, the High Heels and the Low Heels; the war with the neighbouring empire of Blefuscu, in which Gulliver himself, the Man Mountain, secures the victory by carrying off the whole of the enemy's fleet tied by pieces of packthread; the pomp, vanity, and dignity of the little emperor; his reviews of his little army, and his pride in his little palace, the work of so many generations of Lilliputians. In such ways the author shows us the absurdity of our own world, simply by letting us see it all enacted on a smaller scale. In this part, too, it is pretty clear that Swift intended perpetual reference to contemporary events. Lilliput and Blefuscu stand for England and France; the High Heels and Low Heels for Whig and Tory. Bolingbroke and Walpole are frequently in troduced in a manner that at the time must have been unmis-habit, or tendency developed without restraint; even the human takably plain.

In the second part Gulliver, having escaped home from the Lilliputian kingdoms, again sets out on his travels, and again is accidentally left on a strange coast, which proves to be that of Brobdingnag, a land peopled by beings as much larger than Gulliver as he had been than those of Lilliput. Here Swift has

The third part of the book is chiefly taken up by Gulliver's visit to Laputa. And as the first two parts were especially directed against statesmen and politicians, this is mainly directed against philosophers and men of science. In the same part, however, he visits several other strange places, among others Luggnagg, where we meet one of the most powerful and fearful pictures that even Swift has ever drawn, in his account of the "Struldbrugs," or "Immortals," beings endowed with perpetual life, but not with perpetual youth or vigour.

But it is in the fourth part of the "Travels" that the bitter, almost savage spirit of the author, and his contempt for his kind, show themselves in their full strength. Gulliver there visits the land of the Houyhnhnms, a land in which the ruling race are horses, horses raised to a more than human standard of intelligence and cultivation, living in a state of purity, innocence, and simplicity; and having under subjection a race of men turned into brutes, termed Yahoos. In the description of these hideous creatures we can nowhere fail to recognise the human lineaments; but it is humanity with every spark of the higher nature eliminated; every base, low, and sordid passion,

form rendered repulsive and disgusting. We are shown man degraded below the level of the lowest of the brute creation, and placed in deserved subjection to brutes infinitely nobler than himself. And throughout all we cannot but see that Swift intended this not as a mere freak of the fancy, but as a picture of his fellow-creatures.