INTRODUCTION.

SEVERAL of the lectures delivered by Dr. LARDNER in the city of New York were reported for "The New York Tribune," and were afterward published in pamphlet form. The last edition of these lectures was introduced by a "Sketch of the Progress of Physical Science," written by Dr. THOMAS THOMSON, Of London. The publishers of this complete edition of Dr. LARDNER's lectures deem the following extracts from that treatise, respecting the physical sciences of the ancients, an appropriate introduction to these volumes :

The cradle of the human race was beyond dispute the southern portion of Asia-a delightful climate, where the original inhabitants of the earth first lived and multiplied. Chaldea and India had attained a high degree of civilization long before the Greeks and Romans had begun to emerge from a state of barbarism; but we know comparatively little of the attainments in science which these nations had reached. We are equally ignorant of the progress which mathematical and physical inquiries had made in China-not one of the treatises on mathematics, arithmetic, and astronomy, in the Chinese language, having been translated into any of the languages of modern Europe. But the resemblance between the Chinese and the ancient Egyptians is so very striking, and so complete, that it is difficult to avoid suspecting that they had a common origin. If this were so, China, from its contiguity to India and Chaldea, and from the delicious nature of its climate, must have been first furnished with inhabitants. And the Egyptians, if ever they were a colony of Chinese, must have been transplanted into Egypt long before the commencement of history. It was from Egypt that the Greeks drew the first rudiments of their mathematical and physical science; and the scientific acquisitions of that singular people constitute everything that we know respecting the progress which the ancients had made in the investigation of nature.

From the genial climate of the early inhabitants of the east, and the nature of the life which they led, it was natural to expect that the magnificent spectacle of the heavens would speedily attract their attention. We are certain that the Chaldeans made astronomical observations at least as early as the twenty-seventh and twenty-eighth years of the era of Nabonasser; that is to say, seven hundred and nineteen and seven hundred and twenty years before the commencement of the Christian era: for Ptolemy makes use of three observations of the eclipses of the moon, which took place during these years, and which he found in their records. Diogenes Laertius informs us that the Egyptians had preserved in their annals an account of three hundred and seventy-three eclipses of the sun, and eight hundred and thirty-two of the moon, which had happened before the arrival of Alexander the Great in their country. Now these eclipses required between twelve hundred and thirteen hundred years to happen. Alexander's visit to Egypt took place in the year 331 before the Christian era. If we add this number to the length of time during which the Egyptians continued to observe the eclipses of the sun and moon, we obtain sixteen hundred and thirty-one years before the commencement of the Christian era for the period at which the Egyptians began to record their observations. This period is rather more than a century after the death of Moses, and is about twenty-four years before the institution of the Olympic games; constituting but a small part of the forty-eight thousand, eight hundred and sixty-three years during which they boasted that they had been engaged in making astronomical observations; but this was obviously a fable, invented for the purpose of raising themselves in the opinion of the Macedonian conqueror. What progress the Chaldeans and Egyptians had made in astronomy, it is hard to say. They certainly had become acquainted with the planets; but whether the Egyptians had discovered, as Macrobius assures us, that Mercury and Venus revolve round the sun, is not so clear. Their notions respecting the length of the solar year, and the mean length of the lunation, must have been a near approximation to the truth. This is evident from the famous Chaldean period called Saros. It consisted of two hundred and twenty-three lunar months, at the end of which the sun and moon were in the same situation with respect to each other as when the period began. This period includes a certain number of eclipses of each luminary, which are repeated every saros in the same order. foundation of the celebrated Alexandrian school. He collected all the elementary facts known in mathematics before his time, and arranged them in such an admirable order-beginning with a few simple axioms, and deducing from them his demonstrations, every subsequent demonstration depending on and rigidly deduced from those that immediately precede it-that no subsequent writer has been able to produce anything superior or even equal. His "Elements" still continue to be taught in our schools, and could not be dispensed with, unless we were to give up somewhat of that rigor which has been always so much admired in the Greek geometricians. Perhaps, however, we carry this admiration a little too far. The geometrical axioms might be somewhat enlarged, without drawing too much upon the faith of beginners. And were the method followed, considerable progress might be made in mathematics without encountering some of those difficult demonstrations that are apt to damp the ardor of beginners.

The Chaldeans appear to have divided the day into twelve hours, and to have constructed sun-dials for pointing out the hour. The sun-dial of Ahaz is mentioned in the Old Testament, on the occasion of the recovery of Hezekiah; but nothing is said about its construction. Undoubtedly, however, such sun-dials would require a certain knowledge of gnomonics-which, therefore, the Chaldeans must have possessed.

That the Egyptians had made some progress in mathematics admits of no doubt, as the Greeks inform us that they derived their first knowledge of that branch of science from the Egyptian priests. But that the mathematical knowledge of the people could not have been very extensive, is evident from the ecstasy into which Pythagoras was thrown when he discovered that the square of the hypotenuse of a right-angled triangle is equal to the square of the two sides: for ignorance of this very elementary, but important proposition, necessarily implies very little knowledge even of the most elementary parts of mathematics

It was in Greece that pure mathematics first made decided progress. The works of three Greek mathematicians still remain, from which we have obtained information of all or almost all the mathematical knowledge attained by the Greeks. These are Euclid, Appolonius, and

The elements of Euclid consist of thirteen books. In the first four he treats of the properties of lines, parallel lines, angles, triangles, and circles. The fifth and sixth treat of proportions and ratios. The seventh, eighth, ninth, and tenth, treat of numbers. The eleventh and twelfth treat of solids; and the thirteenth of solids: also of certain preliminary propositions about cutting lines in extreme and mean ratio. It is the first four books of Euclid chiefly that are studied by modern geometricians. The rest have been, in a great measure, superseded by more modern improvements.

Appolonius was born at Perga in Pamphylia, about the middle of the second century before the Christian era. Like Euclid, he repaired to Alexandria, and acquired his mathematical knowledge from the successors of that geometrician. The writings of Appolonius were numerous and profound; but it is upon his "Treatise on the Conic Sections," in eight books, that his celebrity as a mathematician chiefly depends.

The conic sections, which, after the circle, are the most important of all curves, were discovered by the mathematicians of the Platonic school; though who the discoverer was is not known. A considerable number of the properties of these curves were gradually developed by the Greek geometricians. And the first four books of Appolonius are a collection