13. An equation is a proposition asserting the equality of two quantities, which is expressed by placing the sign between them; thus 36d. 3s. = = 14. A simple equation is that which contains only one power of the unknown quantity; thus a+b =x is a simple equation. Example. (1.) Suppose x + y = 12 and x - y = 8, to find the value of x and y. x + y = 12 must get rid of one of them by addition or subtraction : I get rid of the y by addition, and find that 2 x = 20 of course x = 10, but x + y = 12, therefore y must be equal to 2. (2.) What two numbers are those whose sum 45 and difference 9, make x and y the unknown numbers then. 1. How are the fundamental operations in algebra performed? 2. How many cases are there in addition, and what are they? 3. What is the rule in addition when the terms are like, and have like signs? Work the examples. How is the operation illustrated? 4. What is the rule when the terms are like, but have unlike signs? Work the examples. How is the operation illustrated? 5. How do you add terms that are unlike? 6. What is the rule for subtraction? Work the examples. 7. What is the rule respecting the signs in multiplication? 8. How do you multiply two terms into one another? Work the examples. 9. How do you multiply compound quantities? Work the examples. Give the explanation. 10. What is division, and what is the rule when the divisor and dividend are simple quantities? Work the examples. 11. What is the rule when the dividend is a compound quantity? Work the examples. 12. What is the rule when the divisor and dividend are both compound quantities? Work the example. 13. What is an equation? 14. What is a simple equation? Work the first example and give the explanation. GEOMETRY. ON MATIIEMATICS. "If a man's wits be wandering let him study mathematics; for in demonstrations, if his wit be called away never so little, he must begin again."-Bacon. THE term mathematics is derived from a Greek word which primarily signifies any discipline or learning. It came in course of time to be limited to one particular description of learning; much in the same manner, and for a similar reason, as the English term learning, has been appropriated to classical knowledge, or the study and acquisition of the dead languages; and it at present denotes that science which teaches or contemplates whatever is capable of being numbered or measured, in so far as computable or measureable; and accordingly is subdivided into arithmetic, which has number for its object, and geometry, which treats of magnitude. "It is not possible," says an eloquent writer in the London Encyclopædia," that a single word should be descriptive of It any complex object presented to the mind, whether that object exist in nature independently of the mind, or be merely one of its own inventions. Hence the term mathematics cannot be taken as a definition, nor can the definitions which have been attempted for it be regarded as any thing more than approximations to accurate indications. If, for example, we say mathematics is that science which contemplates whatever is capable of being numbered or measured, or that it is the science of quantity, or a science that considers magnitudes, either as computable or measureable, all these and similar attempts at definition, are essentially faulty, because they are not sufficiently definite and comprehensive. would perhaps be a nearer approach to accuracy of definition, or rather indication to say, mathematics is the art of computation and measurement. But this also is defective. The term art, however, is rather more appropriate than the term science; though neither the one nor the other is sufficient of itself for the purpose; for both art and science are necessarily included. Mathematics is neither an art nor a science, but the union of both. We are accustomed to hear of mathematical sciences, knowledge, and learning; we never hear, however, of mathematical art; and the expression no doubt seems strange. But this very circumstance proves the importance of our attempt to awaken attention to what may in itself seem of no consequence. We are the unconscious slaves of habit or custom; and the established usage of language is one of the last species of bondage, which even a philosophic understanding completely shakes off." Mathematics are commonly distinguished into pure and speculative, which consider quantity abstractedly; and mixed, which treat of magnitude as subsisting in material bodies, and consequently are interwoven everywhere with physical considerations. Mixed mathematics are very comprehensive, since to them may be referred astronomy, optics, geography, hydrography, hydrostatics, mechanics, fortification, navigation, &c. &c. Pure mathematics have one peculiar and distinguishing advantage, that they occasion no disputes among wrangling disputants, as in other branches of knowledge; and the reason is, because the definitions of the terms are premised, and every body that reads a proposition has the same idea of every part of it. Hence it is easy to put an end to all mathematical controversies, by shewing either that our adversary has not stuck to his definitions, or has not laid down true premises; or else that he has drawn false conclusions from true principles; and in case we are able to do neither of these, we must acknowledge the truth of what he has proved. It is true that in mixed mathematics when we reason mathematically upon physical subjects, we cannot give such just definitions as the geometricians; we must therefore rest content with descriptions, and they will be of the same use as definitions, provided we are consistent with ourselves, and always mean the same thing by those terms we have once explained. The term geometry literally and primarily signifies measuring of the earth, as it was the necessity of measuring the land that first gave occasion to contemplate the principles and rules of this art, which has since been extended to numberless other speculations; insomuch, that together with arith- metic, geometry, as I have already observed, forms now the chief foundation of all mathematics. Herodotus ascribes the invention of geometry to the Egyptians, and asserts that the annual inundations of the Nile gave occasion to it; for those waters bearing away the bounds and landmarks of estates and farms, covering the face of the ground uniformly with mud, the people, he says, were obliged every year to distinguish and lay out their lands by the consideration of their figure and quantity, and thus by experience and habit they formed a method or art which was the origin of geometry. A further contemplation of the draughts of figures of fields, thus laid down and plotted in proportion, might naturally lead them to the discovery of some of their excellent and wonderful properties; which speculation continually improving, the art continually gained ground, and made advances more and more towards perfection. LESSON THE FIRST. 1. Geometry is the science of local extension, as of lines, surfaces, and solids, with that of ratios. 2. The term geometry literally signifies measuring the earth, to which it was first applicable; but it has long since been regarded as the foundation of all mathematical knowledge. 3. The invention of geometry was occasioned by the inundations of the Nile, which every year washed away the boundaries of lands, covered their surface with mud, and obliged the proprietors to lay new claims, by the consideration of the figure and extent of their property. 4. Geometry is distinguished into theoretical and practical. 5. Theoretical or speculative geometry treats of the various properties and relations in magnitudes, demonstrating theorems, &c. 6. Practical geometry is that which applies those speculations to the uses of life in the solution of problems. 7. Elementary geometry is that which is employed in the consideration of right lines and plane surfaces, with the solids generated from them. 8. Geometry depends wholly on definitions and axioms. 9. The definitions in geometry are clear, plain, and universal, such as "A point has neither parts nor magnitude." "A line is length without breadth or thickness." "A surface has length and breadth only." "A solid is any thing that has length, breadth, and thick "An angle is an opening or inclination of two lines meeting in a point." 66 'If one line standing on another, makes the angles on both sides equal, those angles are right angles, and the line standing on the other is a perpendicular to that on which it stan:ls. "A triangle is a plain figure, bounded by three lines or sides." "A circle is a plain figure bounded by a curve line, called the circumference, every part of which is equally distant from a point within, called the centre," &c. &c. 10. An axiom is a manifest truth not requiring any demonstration: as the following 66 Examples." Things equal to the same thing are equal to one another." "The whole is greater than any of its parts, and equal to all its parts." Magnitudes which coincide with one another, or which exactly fill the same space, are equal to one another.” “If equal things are taken from equal things, the remainders are equal." &c. |