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What number is that which being divided by 4, the quotient will be 3? Let x represent the number required.
Then 3 by the question. Now since 3 is the
fourth part of x, x will be 4 times 3, or x = 3 × 4 = 12, the number required. It is obvious that if the unknown number be divided by a given number it may be freed from the divisor by multiplying the other term by that number. The learner cannot fail to perceive that the principles now developed are the same as those which he employed at the very beginning of his scientific studies in proving addition, subtraction, multiplication, and division.
Mathematics is a branch of human knowledge which treats of everything which can be numbered or measured. Arithmetic is that branch of mathematics which teaches us the properties of numbers, and how to calculate or count by means of figures. Geometry is the next great division of mathematics, and treats of every
thing which can be measured by lines, surfaces, or solids. The term mathematics is derived from the Greek, and literally signifies discipline or learning; the reason why the term learning was given to this kind of knowledge, seems to be that the ancients considered arithmetic and geometry of primary importance, as being the foundation of all other branches of science. So firmly were the ancients convinced that a knowledge of geometry was absolutely necessary for the successful cultivation. of all the branches of philosophy, that the motto placed by Plato over the door of his academy, or school, was to this effect: Let no one unacquainted with geometry enter here. This famous school was situated in Academia, a place surrounded with trees near Athens belonging to a person named Academus; hence places where science is cultivated or taught are called Academies. The ancients called this science mathesis, or sometimes mathemata, from which the word mathematics is directly formed. The term geometry literally signifies land-measuring, or the art of measuring land. It is composed of two Greek words, ge the earth or land, and metreo I measure. This science is said to have arisen in Egypt in consequence of the inundations of the Nile destroying the land-marks between different farms or pieces of land. If the story be true, we have a striking example of necessity being the mother of invention; but whether geometry took its rise in Egypt or not, its primary use seems to have been that of land-measuring. This is what might naturally be expected, as men were undoubtedly shepherds and husbandmen before they became architects or sailors. In whatever country geometry took its rise, it was cultivated at a very early period by the Greeks, and hence most of its terms or technical words are derived from the Greek language. You will find the study of geometry one of the most interesting in which you were
ever engaged. The problems on which you will exert your ingenuity are often as curious as the most amusing of what are called Chinese puzzles. Whilst your minds will be thus engaged, you will imperceptibly be acquiring habits of close attention, of correct thinking and reasoning, which you will afterwards find of the greatest use in the active duties of life which you will have to discharge. But it is not merely the training or cultivation of your minds which we propose by the study of this science; a certain degree of acquaintance with its principles is absolutely necessary for understanding every other useful branch of science. Without a knowledge of geometry, you can understand very little of geography, of navigation, of architecture, of engineering, of natural philosophy, of astronomy; in short, a knowledge of the principles of geometry is absolutely necessary for every person who expects to rise above mere manual labour.
There are various ways by which you may acquire the idea of a straight line, besides that given in our text book*. If a fine thread or hair be suspended by one of its ends whilst a weight is hung to the other, it will be stretched into a straight line. If the line be supposed to become finer and finer, till its diameter vanish all together, the idea of a mathematical straight line is obtained. This example will also give us the idea of a stretched line being shorter than an unstretched, or crooked, or curved line. Hence the idea which every person has of a straight line is that it is stretched in the same direction, and that it is the shortest distance from one place to another. You have already an idea of
* Ritchie's Principles of Geometry.
what we mean by a point. If I ask you to make a point at the corner of your paper, you will immediately make a small round dot with the point of the pen. The smaller and neater you make the dot, so much the more will you think you have done what I wished. The dot exists in a particular place or has position, and the smaller its size the more perfect does it determine the point or place required. A point may therefore be considered as the centre of a dot. This language appears very quaint, but it is in reality the same as that of Plato, who calls a point a monad having position in all geometrical constructions: you are to consider the centre of the points of the compasses as geometrical points, and the imaginary line a line joining them as a given straight line. You will now perceive that if I draw a straight line by means of the scale in one direction, and another crossing it, I have determined the position of a point. Again, if I draw another straight line cutting the circumference of a circle, I determine two points. Lastly, if I describe two circles cutting each other, I determine two points, and consequently in each case the position of a straight line.
Before we begin to establish the principles, it may be useful to give you clear ideas of the meaning of the scientific or technical terms which we are to employ. The first is the circle, and its several parts. The word circle is immediately derived from the Latin circulus, a circle or anything circular; this word is the diminutive of the more primary word circus, also signifying anything round, and generally applied to a circle of a considerable size. The word circus comes from a still more ancient stock, namely, from the Greek (kirkos) signifying the same thing. The term circumference is obviously compounded of the two latter words, circum, about or round about, and fero, to carry, and denotes that which is formed by
something carried round. Centre, from centrum, the middle point of any round object. Diameter is from two Greek words, dia, through, and metron, a measure, and signifies the measure through the centre or across the circle. Radius, a Latin word signifying a ray of light from the sun and sometimes the spoke of a wheel. The word arc is derived from arcus, a bow: the word arch formerly used in books on geometry is now confined to architecture. Chord, from corda, a cord or string, in reference to a bow and its string. Degrees, from de, and gradus, a step, and signifies a step. Hence to go by degrees, to go step by step. Minute, a small portion of anything. The term minutes and seconds, signifying small portions of the circumference, are applied also to time, because sixty minutes make an hour and sixty seconds make a minute. The term angle, from angulus, literally signifies a corner, but in geometry it means the inclination of two lines, or rather the opening between them which meet and form a corner. Acute, from acutus, sharp, when applied to an angle, denotes a sharp angle, or one less than 90o. Obtuse, from obtusus, blunt, denotes a blunt angle or one greater than 90o. Parallel, from a Greek word parallelos, signifying equally distant from one another. A figure, from figura, anything having a form or shape. A triangle, from tria, three, and angulus, an angle, a figure having three sides. Equilateral, from æquus, equal, and latus lateris, a side, equal-sided. Isosceles, from isos, equal, and skelos, a leg, signifies having two equal legs or sides. Scalene, from skelos, distorted; a scalenic triangle is therefore a distorted triangle. Parallelogram, from parallelos and gramma, a stroke or letter, and signifies a figure formed by parallel strokes or lines. Rhomboid, rhombus, both from rembein, to sling, and signifies a square or rectangle pulled by the opposite corners, so as to change their