Page images
PDF
EPUB
[graphic][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed]
[subsumed][ocr errors][graphic][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed]
[graphic][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed]
[blocks in formation]

Ir would seem strange that a periodical, supported by those engaged in the pursuit of the higher branches of knowledge, should be entirely without contributions upon scientific subjects. Its contributors cannot, indeed, attempt the style of Euclid or La Place, but there are scientific themes to which they can aspire. Those long versed in the nicest subtilties of the mathematics, find it extremely difficult clearly to unfold the principles of their science to those entirely destitute of their own nice ideas of quantity and ready perception of its relations. And it has been remarked, that modern mathematicians, in particular, exhibit a carelessness in laying down the steps by which they have arrived at their conclusions, strangely discordant with the strictness of the Alexandrian school. It is the humble part of the student to notice the difficulties which he may, in consequence, have encountered.

These considerations have induced us to state the process by which we have endeavored to explain to our own mind certain geometrical principles that have engaged our attention.

Geometry, as its name imports, treats of the dimensions of matter. Founding its reasonings on the perception of the senses, it proceeds, from a few simple and manifest truths, to the most profound conclusions. It is accordingly a very natural error to consider the quantities of which it treats as necessarily material. Philosophy teaches us that matter is impenetrable, and that force can bring matter into space not occupied by other matter. Bodies are said to meet each other in any part, when there is no space between them in that part. Therefore all matter has extension in every direction; for if any body of matter had not extension in any particular direction, two other bodies of matter, on opposite sides of it, could meet in that direction, and there could be no matter between them. Accordingly points, and also lines and surfaces, which are not considered as having extension in the direction perpendicular to them, have only an ideal existence. The same reasoning may be varied so as to show that they are not impenetrablean indispensable idea in conceiving of the coincidence of their seg

[blocks in formation]

ments. This, Simpson seems not to have kept in view, in the demonstration quoted in the second note to Playfair's Euclid.

Not only are points, lines, and surfaces ideal, but they are primitive ideas. Our conceptions of squares, isosceles triangles, &c., are derived and limited from other and simpler ideas; they can be explained to any one who has those primitive ideas, and such an explanation is called a definition. But these primitive ideas cannot be thus referred to others. You might as well talk to the blind man of color, as endeavor to inculcate these ideas by any other method than illustration. Accordingly no definition can be given of them but such as reducing each to its abstract element. Thus a point is well defined as indicating position. But, while it admirably illustrates our introductory remarks, nothing can be more objectionable, as a fundamental definition, than that of Playfair's: "If two lines are such that they cannot coincide in any two points, without coinciding altogether, each of them is called a straight line." Now this is true enough; and so is it true that "the angles at the base of an isosceles triangle are equal to one another;" but the latter he thinks worthy of a labored demonstration. Many a poor wight would have thanked him if, on account of their truth and their clearness to his mind, he had thrown every proposition in his book into the form of a definition. The definition under consideration requires at least a mental demonstration, that some similar curves or broken lines (e. g. the circumference of two equal circles) may not answer its conditions. Not only so, but the preconceived idea of a straight line, the very thing to be defined, is necessary to its demonstration. Again, Playfair's objection to Euclid's definition, that "a straight line is one which lies evenly between its extreme points," is, that the word "evenly" is as much in need of a definition as the thing it would define. It seems to us that there are quite a number of words in the substituted definition which have not been defined, and with regard to which Playfair trusts to the preconceived ideas of men. For instance, it would seem that the word "coincide" is full as vague, to those not familiar with mathematical ideas, as the term "straight." Thus we see the wisdom of the Greek philosopher, in resolving this idea into its abstract element. It would be more in accordance with our modes of thinking and speaking, to say that a straight line is one which nowhere changes its direction.

Again, it is necessary to keep it in mind that these are distinct ideas. They are so blended together that we are apt to forget how totally distinct they are. They all relate to extension; they may all be derived from the same solid; but our ideas of inertia, velocity, &c., are derived from the same. Solids are indeed bounded by surfaces; surfaces by lines; and lines by their extreme points; and so is the momentum of a body determined by its mass and its velocity; yet these ideas are none the less distinct, and, so to speak, incommensurable. We may say that a ratio exists between two lines, and that the same ratio obtains between two surfaces; but no ratio can be established between a line and a surface. Here Geometry, in its too zealous care for its distinctive ideal element, has allowed a seeming advantage to

« PreviousContinue »