success. find the quadrature of the circle, or the ratio of the circumference to the diameter. It is remarkable that the first attempt to solve the most famous problem in Geometry, should have been a prison amusement. Pythagoras was born about 580 years before Christ. After having travelled into Egypt and India, be gave himself up to the study of geometry with wonderful ardour and It was he who discovered that the square of the hypothenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. To express his joy and gratitude for this great discovery, we are told that he sacrificed one hundred oxen to the Muses. He also discovered that the circle is the greatest of all figures of the same perimeter. The first man who digested the Elements of Geometry into a regular treatise, was Hippocrates, who lived soon after Pythagoras. This work has not come down to us; but history informs us, respecting Hippocrates, that he was originally a merchant; that he visited Athens on business, and was one day tempted by mere curiosity to visit the schools of philosophy; that be there heard of geometry for the first time, and was so charmed that he renounced all other pursuits and gave his whole mind to this. No wonder that with such fervent devotion to the study, he soon became one of the best geometers of his time. We now come to the celebrated school of Plato, in which, during the life of its founder, geometry formed the basis of instruction. It is delightful to think of the enthusiasm which so great a man as Plato felt for this study. He placed an inscription over the door of his school, saying, let no one who is ignorant of Geometry enter here." He also declared to his disciples his belief, that the mind of the Deity was constantly occupied with the truths of geometry. For some time the disciples of Plato shared the enthusiasm of their master, and accordingly from them geometry received immense accessions. Leon, a pupil of one of Plato's disciples, arranged, for the second time, the elements of Geometry into a regular treatise. And Eudoxus, an intimate friend of Plato, found out the solidity of a pyramid and It is also supposed that he was the inventor of the theory of geometrical proportion, as presented by Euclid, of whom we are next to speak. About 300 years before Christ, Ptolemy Lagus founded a school of philosophy at Alexandria, in which Mathemat cone. ics was cultivated before everything else. It was heri that Euclid gained his lasting celebrity as a Geometer; his ardour having been first kindled at Athens, under the disciples of Plato. It is related that when Ptolemy Philadelphus asked him, whether there was any easier method of studying geometry than the one commonly pursued, he replied “No : there is no royal road to geometry.” Euclid is chiefly known in modern times as the author of the Elements, a work composed with such wonderful judgment and sagacity, that the efforts of 2000 years have scarcely been able to make an improvement upon it. It has often been re-modelled and has had ever so many commentators; but under some form or other, it is at this day studied in every region of the civilized world. What a glorious earthly immortality did the composition of this work secure to its author ! It is a siugular fact that Euclid's Elements were first known to Europe, after the revival of learning in the 12th century, through the medium of an Arabic translation. Following down the order of time, the next name of celebrity is that of Archimedes, who was born at Syracuse about 287 years before Christ. It was he who first discovered the properties of the sphere and cylinder. Upon these discoveries, he wished his fame with posterity to rest; for which reason, he requested that after his death, a sphere and cylinder might be inscribed on his tomb. But he made a great many other discoveries; and among the rest, that of the approximate ratio of the circumference of the circle to its diameter. He demonstrated that, calling the diameter 1, the circumference is between 37 and 3. and the principles laid down by him in this demonstration, have formed the basis of all succeeding approximations. It is generally admitted that Archimedes holds the same rank among the ancients, as Newton and La Place among the moderns. - The method of Exhaustions described hereafter, was his invention. About the time that Archimedes died, Apollonius was born, a man who acquired such reputation among his contemporaries, as to be familiarly known by the name of the Great Geometer. His writings have fortunately been preserved, and together with those of Euclid and Archimedes, form the chief sources from which our knowledge of an *b vi cient Geometry is derived. After Apollonius, no very distinguished name occurs before the Christian Era. With the Christian Era commences a long interval, in which no brilliant discovery was made. Learning of every kind was now in the wape. Towards the end of the fourth century, two mathematicians appeared, Theon and Pappus, who wrote some excellent commentaries upon former works, but produced nothing original. Hypatia, too, the illustrious daughter of Theon, and his successor in the chair of the Alexandrian school, was famed for her knowledge of geometry and for the sagacity displayed in her annota-. tions upon Apollonius. But these are all who deserve to be mentioned even as commentators, for several centuries. During the fifth, sixth, and seventh centuries, geometry was chiefly cultivated by the Arabs and Persians. The Arabs, without contributing many new discoveries, translated most of the works of the Greek geometers, and by thus preserving the lights of this branch of science from total extinction, made some remuneration to Europe for the general devastation which followed their inroads. The Persians were well acquainted with the Elements of Euclid, and made copious commentaries upon it. One of their most distinguished geometers, Maimon-Reschid, conceived such a singular fondness for one of Euclid's propositions, that he wore the diagram for an ornament embroidered on his sleeve. The Persians call geometry the difficult science, and bave fantastic names for all the principal propositions. For example, they call the proposition respecting the square of the hypothenuse, the bride, and the converse of it the bride's sister: Rome never had any distinguished geometers. Cicero professes a high esteem for Mathematics, but did not write upon the subject. The Chinese have never cultivated geometry to any great extent. When the Europeans first visited them, their knowledge extended little farther than the rules of Mensuration. In Europe from the eighth to the thirteenth century geometry with difficulty maintained a precarious existence. Here and there a solitary individual, in the retirement of a cloister, made it the subject of his contemplations. But on the whole, this period may be properly called the midnight of geometry. During the thirteenth and fourteenth centuries we begin to perceive the dawnings of a brighter day. Among the absurd opinions entertained in the dark ages, one of the most absurd was a belief in Astrology, or in the influence exerted by the positions and motions of the heavenly bodies upon human affairs. Yet to this belief, more than to any thing else, we are indebted for the revival of geometry. The vain attempt to foretell the destiny of an individual, by casting his horoscope, as it was called ; that is, by ascertaining the relative positions of the planets at the time of his birth, led those who professed astrology to an assiduous cultivation of geometry, without which their calculations could not be made. At length, however, as the darkness of ignorance and superstition began to be dissipated, geometry was studied from a nobler motive. Though it takes its name from the measurement of land, yet its noblest application is to the spaces of the heavens. In other words, it forms the key to all our knowledge of Astronomy, by far the most sublime of sciences. With this view, geometry began to be cultivated in the fourteenth and fifteenth centuries. At this time it numbered among its votaries, Wallingfort, the English poet Chaucer, Purbach, and Regiomontanus. But names now begin to thicken upon us in such numbers that we can only mention the most celebrated. Cavalleri was born at Milan in 1598. He invented a new method of geometrical reasoning, called the method of Indivisióles. He considered a line as made up of an infinite number of points, a surface as made up of an infinite number of lines, and a solid as made up of an infinite number of surfaces. These infinitely small elements of the geometric magnitudes, he denominated indivisibles. The method of summing an infinite series of terms in arithmetical progression had long been known; and accordingly the process of comparing curves with straight lines, and measuring the area of surfaces, and the solidities of solids, was now rendered simple and summary. The method of indivisibles has a decided advantage over the ancient method of Exhaustions ascribed to Archimedes, by being far less cumbrous and circuitous. To explain what is meant by the method of exhaustions, we will describe its application to a particular case. Suppose it were required to find the area of a circle. For this purpose, a polygon is inscribed in the circle, and another is circumscribed about it. Here then are two determinate areas, one less and the other greater than that of the circle. Thus two limits are fixed, within which the area sought must be contained; and these limits may be constantly brought nearer together, by increasing the sides of the two polygons. At length the difference between the two limits, is reduced to a quantity too small to be estimated. It is then said to be exhausted, and the area of either of the polygons may be taken for the area of the circle. This is the method which Archimedes employed to find the ratio of the circumference to the diameter. It was also employed by Ludolph Van Ceuben, a Dutch geometer contemporary with Cavalleri for the same purpose. This man had the patience to carry the approximation to 36 figures. Another contemporary of Cavalleri, Roberval of France, invented a method of reasoning which closely resembled the method of indivisibles; but differed in this, that surfaces were considered as made up of an indefinite number of narrow rectangles or oblongs, and solids of an indefinite number of thin prisms, all decreasing according to a certain . law. In the same century Descartes conferred a lasting benefit upon geometry, by applying Algebra to it. By this invention, the properties of geometrical figures are represented by equations; and the Application of Algebra to Geometry, has now become an extensive branch of mathematics. The ancient geometers were entirely ignorant of Algebra, and the discovery of so powerful an instrument, is the most important advantage yet gained by the moderns. In this connexion we must not omit to mention Pascal, especially as we write for youth. Probably France never produced a greater genius. He had heard the mathematicians who visited his father speak with enthusiasm of geometry. He requested that a book of geometry might be given him. This his father refused, because he was yet only twelve years old, and it was not consistent with the plan marked out for his education, that he should commence the study of mathematics so young. But Pascal was not to be thus put off. He had received a hint of what geometry was, and immediately began to invent a system for himself. The walls of his room were literally covered with diagrams, and he had already advanced so far as to demonstrate that the three angles of a triangle are equal |