MATHEMATICS (from the Greek uathiltaTth*. the science which treats of the ratic and comparison of quantities, whence it is defined the science (f ratios ; sonic writers call it the science of quantities ; but this is inacurate, since quan tities themselves are not the subject of mathematical investi gation, but the ratio which such quantities bear to each other.
The term mathenudics is derived from pathiatc, mathcsis, discipline, science, representing with justness and precision, the high idea that we ought to form of this branch of human knowledge. In ffiet, mathematics are a methodical concate nation of principles, reasonings, and conclusions, always accompanied by certainty, as the truth is always evident, an advantage that particularly characterizes accurate knowledge and the true sciences, with which we must be careful not to associate metaphysical notions, nor even the striingest probabilities.
The subjects of mathematics are the comparisons of mag nitude, as numbers, velocity, distance, &c. Thus, geometry considers the relative magnitude and extension of bodies; astronomy, the relative velocities and distances of the planets ; mechanics, the relative powers and force of different machines, &c. &c., some determinate quantity being fixed upon in all cases, as a standard of measure.
The study of mathematics. is highly useful to the architect, particularly arithmetic.geometry, mensuration, and mechanics. Geometry enables him to take his dimensions under the most difficult circumstances, and to lay out the various parts of his design, while it furnishes him with rules for executing the same. Mensuration is the application of arithmetic ti geometry, and shows him how to find the exact of his labour, to the difficult v of executing a certain uniform portion of a work, and to estimate the quantity of materials employed therein : that branch of mathematics called mechanics, enables him to compute the strength and strain of the materials he employs. In short, without the aid of mathematics. he is unfit for his profession ; and the more he understands, the fewer difficulties he will have to encounter in the prosecution of his art.
Mathematics are naturally divided into two classes; the one comprehending what we call pure and abstract ; and the other the coy/yip/no' or mixed. Pure mathematics relate to magnitudes generally, simply, and abstractedly, and are the ref pre founded on the elementary ideas of quantity. Under this class are included arithmetic, or the art of com putation ; geometry, or the science of mensuration and comparison of extensions of every kind ; analysis, or the comparison of magnitudes in general ; to yt hieh we may add geometrical analysis, which is a combination of the two latter. Mixed mathematics are certain parts of !physics, which are, by their nature, susceptible of being submitted to mathe matical investigation. We here borrow Irvin incontestable experiments, or otherwise suppose bodies to possess some principal and necessary quality. and then, by a methodical and demonstratiVe chain of reasoning. deduce from the prin ciples established conclusions as evident and certain as those which pure mathematics draw immediately from axioms and definitions, observing., that these results are always given with reference to the experiments on which they are founded, or the hypothesis Nrhivh furnished the first datum. To illus trate this by an example : numberless experiments have shown us, that all bodies near the earth's surthce fall with an tweelerated velocity. and that the spaces passed through are
as the squares of the time they have occupied in falling. This, then, the mathematician considers as a necessary and essential quality of matter, and with this datum lie proceeds to examine what will be the velocity of a body after any given time, in what time it will have acquired a given velocity. w hat time is necessary for it to have generated a given space, &c., and in all these investigations his conch'. sions are as certain and indisputable as any of those geometry deduces from self-evident truths, and definitions. Again, in optics, having established it as a principle of that it is transmitted in right lines while no obstacle is opposed to the passage of the rays ; that when they be•ome reflected, the angle of incidence is equal to the al gle of reflection ; that in from one medium to another, of ditThrent density, they fly off from their first direction, but still follow a certain geometrical law; these principles, or qualities of light, being once admitted, whatever may be its nature, be it material or ilmnaterial, or may be the medium through which it passes, or the surface by which it is reflected, are matters totally indifferent to the mathematician ; lie considers the rays only as right lines, the sin filers on which they impinge as geometrical planes, of which the form only enters into his investigation : and from this point all his inquiries are purely geometrical. his investigation clear and perspicuous, and his deduction evident and satisfactory. To this class of mathe matics belong mechanics, or the science of equilibrium and motion of solid bodies : hydrodynamics, in which the equili brium and motion of fluids are considered ; astronomy, which relates to the motion, masses, distance, and densities, of the heavenly bu?lies; optics, or the theory and effects of light; and, lastly, acoustics, or the theory of sounds.
Such are the subjects that (idl under the contemplation of the mathematician ; and, as for as a knowledge of these may be considered beneficial to mankind, so far, at least, the utility of the science on which they depend, must be admitted. It isnot, however, the application of mathematics to the various purposes of society, that constitutes their particular excellency ; it is their operation upon the mind, the vigour they impart to our intellectual faculties, and the discipline which they impose upon our wandering reason. "The mathematics," says Dr. Barrow, "effectually exercise, not vainly delude, nor vexatiously torment studious minds with obscure sub tilties, but plainly demonstrate everything m ithin their reach, draw certain conclusions, instruct by profitable rules, and unfold pleasant questions. These disciplines also inure and corroborate the mind to a constant diligence in study ; they wholly deliver us from a credulous simplicity, and most strongly fortify us against the vanity of scepticism ; they eactually restrain us from a rash presumption, most easily incline us to a due assent, and perfectly subject us to the government of right reason. Whde the mind is abstracted and elevated from sensible matter, it 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 finicy calmed and settled, and the understanding raised and excited to more divine contemplations."