AUTHOR OF THE "CHART OF GEOMETRY" AND "FIRST LESSONS." NEW YORK: COLLINS, BROTHER & CO., NO. 254 PEARL STREET. 1846. Entered, according to Act of Congress, in the year 1846, By D. M'CURDY, In the Clerk's Office of the District Court of the Southern District of New York. OR-2-41 WEL 11-28-41 44358 PREFACE. To bring the Elements of Geometry into general use, is the design of this volume, and of the "Chart of Geometry" and "First Lessons" which precede it. There was once a competition between certain persons to be the first who should see the risen sun; and the prize was awarded to him who turned his face westward: because there the sun's effects were first discovered, in gilding towers, and battlements, and the mountain's brow. To ascertain the existence of geometry by its effects, let us turn from books to the community, and the obvious defect will meet us in every department of life. Few citizens know what these things mean, or what their use. The A question then arises, "Should this be so?" regrets of thousands prove the contrary. The learning to read and write is a mere preparation to receive instruction: after which, the learner should take hold of the properties of things, and examine them in detail, beginning with the most general, and therefore the most useful. But are there any properties more general than those of magnitude, figure, and motion? There are none: the attribute of number itself is not more general, and it is certainly less expedient as a branch of study. The cherished motto, "A place for everything," evinces the necessity of geometry in all the schools. The magnitude and figure of everything, and of the space to contain it, as well as the law of motion and the momentum of force which conveys it to the place, are certainly more worthy of consideration than the mere fact that it counts one. It is obvious from the perfection in which the elements of geometry have been handed down to us, that the Greeks taught these elements in all their schools; that geometry was to them what arithmetic has been to us, namely, the groundwork of public instruction. See, then, the effects of this practice in their works of art, their architecture, their sculpture, their literature, their philosophy, the spread of their language, the respect paid to them by the Romans after the conquest of Macedonia. The advantages of a right education to a people are incalculable. An opinion is widely entertained, namely, that algebra should have priority of geometry in the order of study. This reverses the natural order of the studies; for what is algebra but a method of managing arithmetic and geometry? It prepares certain general formulæ, and teaches the reduction of them, by transposition, substitution, elimination, &c. This may be done as an envelope is prepared for a letter:-the letter must be enclosed, or the envelope is of no use. It requires knowledge of the relations of numbers, of magnitudes, or figures, as the case may be, to dispose in an equation the data and quæsita of a proposition. Any simple problem will illustrate the absoÎute dependence of algebra upon geometry, and how preposterous the idea of giving it priority. Let us seek the ordinate of a circle from its relation to the diameter (d) and the abscissa (x): the formula is this, √dx-x2-ordinate. Now, in view of the diagram, these relations are plainly scen; but without it, the formula would be as abstruse to the juvenile capacity as the Chinese language. After this manner, a multitude of useful theorems are lost to the community, from two causes: one is the neglect of geometry; the other is, that the theorems are placed under the unfriendly umbrage of algebraic symbols; and this latter calls itself the "modern improvements of science!" But that is no improvement in science which precludes the general diffusion of knowledge. There is therefore a want of order, as well as a defect, in the parts which constitute the elementary education. Mathematical reasoning is conducted according to two methods; one is called the method of analysis, or resolution; the other is called the method of synthesis, or composition. Algebra adopts the former of these; separating the known from the unknown parts of a general proposition; representing number and magnitude by symbols, and descending, by a succession of equivalent propositions, from the most complex to the simplest form. Geometry adopts the synthetic method, which begins at the simplest elements, and proceeds, by easy steps, to the more complex combinations; and it is proper to remark, that this alone is the process by which the known and the unknown parts of a general proposition can be distinguished. It is obvious that the cost of books, on this plan, and the labor of the study, are both greatly reduced; and the method by previous recitations of the text, which is the exclusive object of the book of "First Lessons,” and which is continued in the present volume, will enable teachers less proficient to use the work, without apprehension of error or loss of time. There is, moreover, an assurance of success connected with this system, which no other has given, or can give. What is the deficiency with all of us? Why have we not more men entitled to degrees? The reason is this: We are not masters of the elements of science; we cannot call up at will the proofs of many propositions: we wanted this culture when we were a young people; we want it still, and should take heed lest we entail the same want on our posterity. In allusion to the peculiarities of this work, it is unnecessary to be specific. Let the book be examined on its merits, and candidly compared with volumes of two or three times the size; no defect, it is believed, will be found in it, and it will not seem to be redundant. The order of Euclid has been preserved, because it has never been excelled; but the repetitions which swell other editions and perplex the learner, are here obviated by the plan of previous recitations. The demonstrations of the fifth book have been simplified exceedingly; and in one or two instances reduced from two octavo pages to a few lines: not by the substitution of symbols for words, but by a new definition of ratio, and a slight alteration in the method of compounding ratios. Nevertheless, care has been taken to introduce nothing which could not be directly employed in drawing out the properties of proportionals according to the rigor of the Euclidian geometry. The change consists in fixing definitely the value of ratio agreeably to its general use in the mathematics. Mental arithmetic and mental algebra are sublimated abstractions, not affording the proper exercise for the juvenile mind. Children are entitled to the use of their senses before they are required to reason: because, to reason is to combine, compare, digest, and dispose in order the materials received through the senses. Now, arithmetic furnishes an infinite variety of series, the terms of which have varied relations to each other: |