tween them. He observed, by thus increasing the number of sides for a great number of times successively, that he approached a certain assignable rectilinear area, and could come nearer to it than any difference how small soever. It was evident that this rectilinear area was the real size of the curvilinear area to be measured. It was in this way that he found that two thirds the rectangle under the abscissa and ordinate of a parabola, is equal to the area contained by the abscissa and ordinate, and that part of the circumference of the parabola lying between them. In the same way he obtained an approximate measure of the area of the circle, demonstrating that if the radius be unity, the circumference is less than 378, and greater than 341. His two books on the sphere and cylinder were conducted by a similar method of reasoning. He measures the surface and solidity of these bodies, and terminates his treatise by demonstrating that the sphere (both in surface and solidity) is two thirds of the circumscribed cylinder.

In the same spirit his "Treatise on Conoids and Spheroids" was conducted. These names he gave to solids formed by the revolutions of the conic sections round their axis. We pass over his researches on the "Spiral of Archimedes," as it is usually called, though in reality discovered by Conon, one of his friends; but must notice the treatise entitled "Psammites," or "Arenarius." Some persons had affirmed that no number, however great, was sufficient to express the number of grains of sand situated on the seashore. This induced Archimedes to write his treatise, in which he demonstrated that the fiftieth term of a ducuple increasing progression is more than sufficient to express all the grains of sand contained in a sphere, having for its diameter the distance between the earth and the sun, and totally filled with grains of sand. The treatise is short, but abstruse, in consequence of its imperfect method of expressing numbers employed by the Greeks. Were our figures substituted for the Greeks letters, the reasoning would be sufficiently simple and clear.

Archimedes did not confine himself to pure mathematics: he turned his attention likewise to mechanics, and may in some measure be considered as the founder of that important branch of physical science. He first laid down the true principles of statics and hydrostatics. The for

of the most celebrated. It has been noticed by grave philosophers, and sung by poets, as may be seen in the following epigram of Claudian :

"Jupiter, in parvo cum cerneret æthera vitro,
Risit, et ad superos talia verba dedit:
Huccine mortalis progressa potentia curæ ;
Ecce Syracusii ludimur arte senis."

Archimedes wrote a description of this machine, under the name of "Sphæropæia;" but it is lost, and with it everything respecting the nature of the sphere has perished.

The burning mirrors, by which he is said to have set fire to the Roman vessels in the harbor of Syracuse, were long considered as fabulous. But Buffon showed how, by placing a number of small mirrors so that every one of them should reflect the image of the sun to the same point, heat enough might be produced to kindle wood at the distance of one hundred and forty feet.

The protracted defence of Syracuse against the Romans, chiefly in consequence of the wonderful mechanical inventions of Archimedes, is too well known to be enlarged on here.

If we except the discoveries of Archimedes in statics and hydrostatics, hardly any other branch of physical science was much cultivated by the ancients. They have made, indeed, considerable progress in the knowledge of acoustics, so far as music is concerned. In optics they can scarcely be said to have made any progress of consequence; and, in astronomy, very little till the time of Hipparchus, who may be considered as, in some measure, the founder of that sublime science.

Dr. Thomson lays down two methods by which the physical sciences are advanced: observation and experiment; and the application of mathematical reasoning to deduce new facts from principles already established. We give his remarks on observation and experiment, in which he exhibits an analysis of the theory of Bacon on this subject:

It was not to be expected that mankind should at first make any rapid progress in investigating the laws which regulate the changes that take place in the material world. The objects were too numerous and too varied, and escaped his attention by their very regularity. Everywhere in the early ages of the world, we meet with descriptions of prodigies

and wonders, while the regular operations of nature scarcely attracted attention. The method of investigating nature by observation and experiment was scarcely thought of, except by two individuals, who, by means of them, made some progress in mechanics and hydrostatics, and in astronomy: these were Archimedes and Hipparchus. The mechanical discoveries of Archimedes were slightly extended by Ctesibius and Hero, by Anthemius, and by Pappus; while the astronomical observations begun by Hipparchus were continued by Ptolemy.

But at the revival of letters, in the sixteenth century, a spirit of observation and inquiry awoke, which nothing could damp, and men began to pry into the secrets of nature, by the way of experiment. Galileo, in Italy, and Gilbert, in England, especially the former, constitute remarkable examples of the successful investigation by experiment. But it was Francis Bacon, Lord Verulam, who first investigated the laws according to which such experimental investigations should be conducted, who pointed out the necessity of following these laws in all attempts to extend the physical sciences, and who foretold the brilliant success that would one day repay those who should adopt the methods which he pointed out. This he did in his "Novum Organum," published in the early part of the seventeenth century.

Before laying down the rules to be followed in his new, or inductive process, Bacon enumerated the causes of error, which he divided into four sets, and distinguished, according to the fashion of the times, by the following fanciful but expressive names:

Idols of the tribe;
Idols of the den;

Idols of the forum;

Idols of the theatre.

The idols of the tribe are the causes of error, founded on human nature in general. Thus all men have a propensity to find in nature a greater degree of order, simplicity, and regularity, than is actually indicated by observation. This propensity, usually distinguished by the title of spirit of system, is one of the greatest enemies to its progress that science has to struggle with.

The idols of the den are those that spring from the peculiar character

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of the individual. Each individual, according to Bacon, has his own dark cave or den, into which the light is imperfectly admitted, and in the obscurity of which an idol lurks, at whose shrine the truth is often sacrificed. Some minds are best adapted to catch the differences, others the resemblances of things. Some proceed too rapidly, others too slowly. Almost every person has acquired a partiality for some branch of science, to which he is prone to fashion and force every other.

The idols of the forum are those which arise out of the intercourse of society, and especially from language, by means of which men communicate with each other. It is well known that words, in some measure, govern thought, and that we cannot think accurately unless we are able to express ourselves accurately. The same word does not convey the same idea to different persons. Hence many disputes are merely verbal, though the disputants may not be aware of the circumstance.

The idols of the theatre are the deceptions which have taken their rise from the systems of different schools of philosophy. These errors affected the philosophy of the ancients more than that of the moderns. But they are not yet without their effect, and often act powerfully upon individuals without their being aware of their effect.

After an historical view of science from its dawn among the Greeks to his own time, and pointing out the little progress which it had made, in consequence of the improper way in which it had been cultivated, Bacon proceeds, in his second book, to point out the true way of advancing science by induction.

The first object ought to be, to prepare a history of the phenomena to be explained, in all their modifications and varieties. This history is to comprehend not only all such facts as spontaneously offer themselves, but all the experiments instituted for the sake of discovery, or for any of the purposes of the useful arts. It ought to be composed with great care; the facts should be accurately related and distinctly arranged their authenticity carefully ascertained, and those that are doubtful should be marked as uncertain, with the grounds for the judgment formed. This record of facts Bacon calls natural history.

The next object is, a comparison of the different facts, to find out the cause of the phenomenon.