Addition and Subtraction of Surds Multiplication and Division of Surds INTRODUCTION. THE word MATHEMATICS, like most other of our scientific terms, is of Grecian origin, and seems originally to have signified knowledge, or learning in general. It is now, however, greatly limited in its signification, being applied to designate that science exclusively which treats of quantity. Quantity may be contemplated under two different forms; either as made up of separate and distinct parts, or as one extended and continuous whole. When quantity is considered as a collection of separate and distinct parts, or as an ag gregate of several things of the same kind, it is called Number; and that portion of Mathematics which proposes to investigate the power and properties of number is denominated ARITHMETIC. When quantity is considered as one extended and continuous whole, such as in the case of a line, a surface, or a solid, it is called extension; and that division of Mathematics, which proposes to investigate the properties of figured extension, is denominated GEOMETRY. Algebra, another very important branch of Mathematics, seems to hold a kind of middle ground between Arithmetic and Geometry; possessing several of the distinctive qualities of each, yet the exclusive characteristic of neither. For, although it is closely allied to the former, in regard to the manner of its operations, yet it is equally so to the latter, in regard to the generality of its conclusions. Hence, it has been sometimes very appropriately called UNIVERSAL ARITHMETIC. It serves as that link by which Geometry is connected with Arithmetic; or that medium through which Geometrical relation may be exhibited in number. A When quantity is considered in the abstract, that is, apart from all considerations of its existing in any material object whatever, the science is called PURE MATHEMATICS. When quantity is considered as not abstracted from the subject of its existence, the science is called MIXED MATHEMATICS. That quantity cannot exist, unless in some Physical object, is certainly very evident; it is nevertheless convenient and philosophical, by a process of mental abstraction to conceive of it separately. By this means we become armed with a series of speculative truths, which enable us to push our way through new and untried difficulties; to overcome every obstruction, however formidable, by which truth may be encompassed, and to proceed with a firm and a sure step in all our scientific investigations. The study of Mathematics, whether as to their elegance or utility, cannot be better recommended than in the words of Dr Barrow, in his inaugural oration when appointed professor of Mathematics at Cambridge. "The Mathematics, he observes, effectually exercise, not vainly delude, nor vexatiously torment studious minds with obscure subtleties, but plainly demonstrate every thing within their reach, draw certain conclusions, instruct by profitable rules, and unfold pleasant questions. These disciplines likewise inure and corroborate the mind to a constant diligence in study; they wholly deliver us from a credulous simplicity, most strongly fortify us against the vanity of scepticism, effectually restrain us from a rash presumption, most easily incline us to a due assent, and perfectly subject us to the government of right reason. While the mind is abstracted and elevated from sensible matter, distinctly views pure forms, conceives the beauty of ideas, and investigates the harmony of proportions, the manners themselves are sensibly corrected and improved,-the affections composed and rectified, the fancy calmed and settled, and the understanding raised and excited to more divine contemplations." The fundamental principles of mathematics are so few and so easily apprehended-the chain of reasoning so complete and satisfactory-the truths established so clear and perspicuous—the relations discovered so replete with elegance and beauty-that, whilst the reasoning faculties are insensibly gaining strength by the most delightful exercise, the mind itself becomes more and more gratified by the clearness of its views, by the certainty of its conclusions, and by the justness and distinctness of its conceptions. But, besides that mathematics may thus be considered as one of the best exercises to the understanding, they merit our most scrupulous attention, even from equally important considerations; inasmuch as they lie at the very basis of our finest commercial speculations; are intimately connected with the various branches of Mechanical Philosophy; with almost all the great sources of national wealth and power; and with many of those arts which contribute to the elegance and comfort of social life. PART I. ARITHMETIC. If 1. ANY single object whatever is called a unit or one. to this unit another be added, these together form the number called two; which increased in like manner by another unit, gives the next greater number denominated three. By continuing to add unity in this way to the number already obtained, the successive numbers four, five, six, &c. are formed. From this method of conceiving the formation of number, it is obvious that there is no limit to its magnitude; for, however great any number may be, it may still be augmented by a unit, or by any number of units. 2. In the early stages of society, when words, the signs of ideas, were but few, mankind would naturally represent any small number by some palpable symbols, such as nuts, pebbles, or shells arranged in a row. It would be found, however, when the objects to be represented were numerous, that this method of arranging them in a single row would only convey a very confused idea of multitude. In order, therefore, to greater precision, they would subsequently form them into two rows, or count them off by pairs. Hence the origin of the dual number, and of the terms brace, couple, &c. Thirty-one, for example, might be represented by as many small shells or pebbles, which, when ranged into successive pairs, would amount to fifteen pairs and one over, thus: If a pebble or shell of double the size were now taken to represent each pair, then the number, by the same process of arrangement, would be represented by seven pairs of these double-sized counters and one over, to gether with one counter of the first size, thus: When quantity is considered in the abstract, that is, apart from all considerations of its existing in any material object whatever, the science is called PURE MATHEMATICS. When quantity is considered as not abstracted from the subject of its existence, the science is called MIXED MATHEMATICS. That quantity cannot exist, unless in some Physical object, is certainly very evident; it is nevertheless convenient and philosophical, by a process of mental abstraction to conceive of it separately. By this means we become armed with a series of speculative truths, which enable us to push our way through new and untried difficulties; to overcome every obstruction, however formidable, by which truth may be encompassed, and to proceed with a firm and a sure step in all our scientific investigations. The study of Mathematics, whether as to their elegance or utility, cannot be better recommended than in the words of Dr Barrow, in his inaugural oration when appointed professor of Mathematics at Cambridge. "The Mathematics, he observes, effectually exercise, not vainly delude, nor vexatiously torment studious minds with obscure subtleties, but plainly demonstrate every thing within their reach, draw certain conclusions, instruct by profitable rules, and unfold pleasant questions. These disciplines likewise inure and corroborate the mind to a constant diligence in study; they wholly deliver us from a credulous simplicity, most strongly fortify us against the vanity of scepticism, effectually restrain us from a rash presumption, most easily incline us to a due assent, and perfectly subject us to the government of right reason. While the mind is abstracted and elevated from sensible matter, distinctly views pure forms, conceives the beauty of ideas, and investigates the harmony of proportions, the manners themselves are sensibly corrected and improved,-the affections composed and rectified, the fancy calmed and settled, and the understanding raised and excited to more divine contemplations." The fundamental principles of mathematics are so few and so easily apprehended-the chain of reasoning so complete and satisfactory-the truths established so clear and perspicuous-the relations discovered so replete with elegance and beauty-that, whilst the reasoning faculties are insensibly gaining strength by the most delightful exercise, the mind itself becomes more and more gratified by the clearness of its views, by the certainty of its conclusions, and by the justness and distinctness of its conceptions. But, besides that mathematics may thus be considered as one of the best exercises to the |