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capacious intellect be lost in the speculation. Take, for example, the first man, endeavour to depict his thoughts or his feelings, on the first day or the first night, and you will find it exceedingly difficult, even if you can reach his probable conceptions, to select adequate language in which to describe them to others.

Our great poet, Milton, seems to have been aware of this difficulty; and in many parts of his matchless production, Paradise Lost, has evidently restrained his imagination for the purpose of avoiding those incongruities upon which he might otherwise have fallen. Yet, he has not escaped them altogether. He fancies both the first man and his angelic instructor to employ language, which is obviously inappropriate, insomuch as it implies an acquaintance with practices, customs, and ceremonies, which have only found place in a maturer state of the world.

Adam speaks of

"The pledge of day, that crown'st the smiling morn:"
"The sun" that "paints their Heecy skirts with gold."

Raphael, in describing to him the order of creation, speaks of light being

"Transplanted from her cloudy shrine, and plac'd
"In the sun's orb:"

of the

"Great palace noV of light:"

of the stars, which

"In their golden urns draw light:" of the morning planet, which

"Gilds her horns:"

of the

"Glorious lamp," seen in the east, > "Regent of day."

And even in that exquisite passage, in which, with such a noble mixture of genuine philosophy and touching poetry, there is an attempt

"For man to tell how human life began:" the whole is, in my judgment, faultless, at least in reference to this particular, till the new-created mortal, tired with his own emotion, overwhelmed with his own delight,

"On a green shady bank, profuse of flowers,"
Sits " pensive down," where " gentle sleep
First found" him, and" with soft oppression seiz'd"
His " drowsed sense.

Who does not perceive in all this, that the poet, in spite of his obvious efforts to the contrary, blends the circumstances of the pristine state with those of the times in which he lived; — and thus makes the first man, while he was yet without progeny, and his celestial instructor, illustrate surrounding objects and the emotions they excite, by metaphors drawn from an acquaintance (to him impossible) with the arts, practices, institutions, benefits, and evils of polished society ;—with pledges, substitutions, painting, gilding, lamps, urns, shrines, crowns, monarchs, regents, freedom, tyranny, and oppression? Be it observed, however, that I advert to these palpable incongruities, not with the intention of censuring this admirable poet, but of evincing the nature of the difficulty, which so powerful an intellect, so sublime a genius, as his, though fully conscious of it, was unable to surmount.

In tracing the progress of science, the difficulty is of a different kind. We are not so often at a loss to ascertain what was early known with regard to any department of scientific research, or to describe it to others, when ascertained, as to mark the chronological steps in the series, to show the gradations by which it may have passed from its rude, or accidental, or imperfect origin, to the mature state in which we now behold it, and from which we are incessantly deriving so many advantages. Much, it is true, has been done in this respect, but much more remains to be accomplished. The simile of a bridge of numerous arches, by which human life has been sometimes aptly illustrated by the moralists, might with a simple inversion be applicable to what is before us: there the whole of the bridge is seen, except its extremities, which are pictured as enveloped in clouds; here the extremities are illuminated, while mists and fogs hang over many portions of the intervening space. It may, however, be useful, especially to the younger votaries of philosophy, to fix the attention for a while upon the extremities which are most plainly marked,—to contrast the appearance of the early germ with that of the mature plant, —to meditate upon the astonishing difference between the first thought and the expanded series of deductions from it, between the naked, insulated propositions which were first educated, and the complete system of acknowledged verities, of which they at present form perhaps a very inconsiderable part. Such is the object to which I would now direct your attention: and, as I am unwilling to draw too largely upon your candor and forbearance, I shall endeavour to make appropriate selections; referring, for a fuller draught from so rich a stream, to the historians of science.

Allow me to commence with the subject of pure Mathematics. Among its earliest promoters was the celebrated Pythagoras, author of the appellation philosopher,1 and "rendered immortal in the annals of geometry, " say the historians, by the invention of the multiplication table, and by the discovery of three propositions, viz. That only three regular plane figures, the equilateral triangle, the square, and the hexagon, can fill up the space about a point. That the sum of the three angles of every plane triangle, is equal to two right angles. And, that in any right-angled plane triangle, the square on the longest side, is equal to the sum of the squares on the two other sides. The discovery of this last proposition, excited in the mind of Pythagoras such ecstatic and devout feeling, that he is described as offering a hecatomb to the gods on account of it. This I am inclined to disbelieve, for the reason assigned by Cicero, " that it was inconsistent with his principles, which forbad bloody sacrifices." But, if the story be, as is probable, a mere fiction, it still serves to mark the state of mathematical knowledge, when a proposition which, however fertile in its consequences, is now placed amongst the lowest elements, should be characterised as the most brilliant discovery of the great man to whom we owe it.

It does not comport with the nature of an address like the present, to contrast these in detail with the modern state of geometry, of algebra, of fluxions, of the trigonometrical analysis, of logarithms, and exponentials, of series, of rectifications, quadratures, cubatures, tangencies, points of contrary flexure; together with the sublime researches in the theories of isoperimeters, variations, and partial differences. Much less can I attempt to sketch the diversified applications of these to mixed mathematics, and the contributions which have thus been made to the arts and commodities of civilised life. On these topics it would, in truth, be delightful to expatiate, were it not that I should in consequence be compelled to exclude others on which I am desirous of presenting a few remarks. It will, however, be acknowledged by all competent judges, that the eulogium of Dr. Barrow, though elaborate, is not extravagant,

1 While Pythagoras was at Phlias, Leon, the chief of the JPhliasians, was exceedingly charmed with the ingenuity and eloquence with which he discoursed upon various topics, and enquired of him in what art he principally excelled: to which Pythagoras replied, that he did not profess himself Maiter of any art, but that he was a philosopher. Struck with the novelty of the term, Leon asked Pythagoras, who were philosophers, and in what they differed from other men. Pythagoras, in reply, observed that, as at the public games, whilst some are contending for glory, and others sell their wares in pursuit of gain, there is always a third class, who attend merely as spectators; so, in human life, amidst the various characters of men, there is a select number, who despise all other pursuits, and assiduously apply themselves to the study of nature and the search after wisdom; these, added Pythagoras, are the persons whom I call Philosophers. Cic. Tuscul. l>isp. lib. v. c. 3.

especially when applied to the investigations and discoveries of modern times. This great man, with the richness of expression which distinguished his scientific as well as his theological productions, speaks of the mathematics as that " which effectually exercises, not vainly deludes, nor vexatiously torments studious minds with obscure subtleties, perplexed difficulties, or contentious disquisitions; which overcomes without opposition, triumphs without pomp, compels without force, and rules absolutely without causing the loss of liberty: which does not privately over-reach a weak faith, but openly assaults an armed reason, obtains a total victory, and puts on inevitable chains"which plainly demonstrates and readily performs all things within its verge." "The mathematics, which depends upon principles clear to the mind, and agreeable to experience, and draws irresistible conclusions:" "Which is the fruitful parent of, I had almost said, all arts, the foundation of sciences, and the plentiful fountain of advantage to human affairs. In which last respect we may be said to receive from the mathematics the principal delights, securities, and conveniencies of life." To this it is principally owing, as a theoretic spring of action and benefit, "that we dwell elegantly and commodiously, build convenient houses for ourselves, erect stately temples to God, and leave wonderful monuments to posterity:" "That we have safe traffic through the deceitful billows, pass in a direct road through the trackless ways of the sea, and arrive at the designed ports by the uncertain impulse of the winds: that we rightly cast up our accounts, dispose, tabulate, and calculate scattered ranks of numbers, and easily compute them, though expressive of heaps of sand or mountains of atoms:—that we make pacific separations of the bounds of lands, examine the moments of weights in an equal balance, and distribute every one his own by a just measure:—that with a light touch we thrust forward ponderous bodies which way we will, and stop a huge resistance with a very small force :—that we accurately delineate the face of this terraqueous globe, and by diagrams subject the economy of the universe to our sight:— That we aptly digest the flowing series of time, distinguish what occurs by appropriate intervals; rightly predict and discern the various returns of the seasons, the stated periods of years and months, the alternate increments of days and nights, the doubtful limits of light and shadow, and the exact differences of hours and minutes: —that we derive the subtile virtue of the solar rays to our uses, infinitely extend the sphere of sight, enlarge the near appearances of things, bring to hand things remote, discover things hidden, search nature out of her concealments, and unfold her mysteries M freely range through the celestial fields, measure the magnitudes and determine the intervals of the stars, trace the inviolable laws, of their motions,' and ascertain the limits within which the wandering circuits of the heavenly bodies are confined." From this inexhaustible general topic, let us descend to a few particulars drawn from several departments of science; taking first, that of the pressure and motion of liquids. Every one who hears me, will recollect the story "of Archimedes's ecstacy, when he discovered the principle by which might be detected the fraud of the goldsmith who made Hiero's crown. An accidental thhougt, suggested while he was bathing, led to the investigation of a series of propositions, enunciated and demonstrated in his two books "De Insidentibus inJluido" now extant. The nine propositions demonstrated in the first book, relate to the principal laws and circumstances of bodies floating on liquids, or sinking or rising in them; such as are now placed in the elementary enquiries respecting specific gravity. The second book comprises ten propositions, serving to ascertain the different positions which would be assumed by a parabolic conoid in a liquid; according to the different relations of the axis to the parameter, and to those of the specific gravities of the conoid and the liquid. The strictness and elegance of the demonstrations accord with the highest conceptions that can be formed, of the powers of that extraordinary philosopher; but they relate to truthswhich now constitute the amusements of school-boys. Contrast them for a moment with the present state of Hydrostatics and Hydrodynamics; with the fine chain of researches which relate to the equilibrium and pressure of liquids and fluids, whether simple or mixed,—the stability of floating bodies,1— their motions and oscillations when the equilibrium is disturbed, —the general principles which determine the operation of liquids in motion, or of liquids affected by solid bodies in motion, —the operation of pumps,—the discharge of liquids through orifices in the bottoms or sides of vessels,—the motion of water in rivers, canals, open or closed pipes,—the construction of flood-gates, sluices, dams, and banks,—the oscillatory motion of liquids in syphons and other tubes,—the percussion and resistance of liquids, with their application to the motion of water-wheels of different kinds, to the structure and manoeuvres

1 Barrow, Prefatory Oration to his Mathematical Lectures. 2 Among the investigations on this subject, so interesting to those who engage in naval architecture, it would be unjust not to mention, besides those of Bouguer, Euler, Atwood, and Juan, those of M. Charles Dupin, a young French mathematician of considerable genius. His researches on the stability of vessels exhibit a singular union of the strictness of a geometer, the elegance of a classic, and the inventive fancy of a poet. Propositions which before required the attention of able mathematicians, are through the ingenuity of this author, while they are accurately demonstrated, brought down to comprehension of the student.

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