The sciences classed as pure mathematics are Arithmetic, Geometry, Algebra, Analytical Geometry and Calculus. Arithmetic is eminently the science of numbers, and treats of, or practically illustrates their nature and uses. It employs the nine Arabic digits or figures with the addition of the cipher, giving them various positions to express numerical values, and not the native qualities or functions of the things to which they are applied. The methods are the same, and the results obtained equally true, whatever may be the nature of the quantities about which inquiry is made. The elementary or fundamental idea in arithmetic is unity, expressed by the figure 1, from which, with the help of the other eight digits, and the individually valueless cipher, 0, expressions for all the other values, whole or fractional, are formed. As arithmetical processes underlie, or enter into, the work of nearly all mathematical calculations, its great importance as a science is evident; though as often taught in our schools and used in business, it is simply a method of reckoning or computation. Algebra is a kindred science, that, by the use of letters and symbols, enables us to solve more readily all difficult questions relating to numbers. It is, indeed, a kind of universal arithmetic. In the ordinary arithmetic the numbers or figures employed, taken separately, have always the same value, and the result, when, sometimes by a tedious process, obtained, is applicable only to the particular question proposed, but in solving the problem by algebra, since we employ letters to which any values may be attributed at pleasure, the result obtained is largely applicable to all questions of a particular class. Thus, having the sum and difference of two quantities given, we readily obtain an algebraic expression for the quantities themselves. By the new method the goal is reached speedily, and the cabalistic terms, that may, at his first attempts, perplex and discourage the young student, become his delight; and in many difficult processes greatly shorten the work, enabling him with ease to solve problems that to the common arithmetician are tedious, if not impossible. Geometry, one of the oldest of sciences, measures extension, treats of order and proportion in space. Its working elements are not numbers or symbols, but points, and lines, either straight or curved, and surfaces, with volumes, or solids. The simpler problems, when successfully demonstrated, are used in solving those more complicated, making the progress easy.