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means by a plane superficies,' or more shortly
plane.' So of the 'line.' If a leaf of paper be folded and divided into two equal parts by creasing it down the middle, the crease will represent a straight line. Now such a crease does not belong to either half of the paper, but simply divides one half from the other. Further, although it has appreciable width, we can imagine these two halves of the leaf to meet in the middle of the crease, thus making at their junction a line without width running from the top of the leaf to the bottom. Thus we form a notion of Euclid's line, which is "length without breadth," and it is this mental conception, and not our imperfect representations of it on paper, which Euclid invariably has in view when he uses the word 'line.' Having thus conceived of a line, we can fix our attention on its ends and form the
notion of a 'point.' The line has no thickness, no width. The point has no length, and is therefore "that which has no parts or which has no magnitude.” What is here said of the plane, line, and point is equally true of all the objects mentioned in the definitions. They are purely mental conceptions. Thus our representations of lines, angles, triangles, circles, &c., are all imperfect, as are also the concrete examples of these which we see around us, and all that is said of them in the definitions, or afterwards proved concerning them in the propositions, is true of these mental conceptions and of these only. For this reason geometry is called an abstract science.' A further point to be noticed in regard to the definitions is the relation in which they stand to the whole system of geometry. It has been already remarked that they form the groundwork of
the entire subject. They supply the first principles of the science, and the elementary conceptions from which all subsequent truths are derived. It will be noted that they are arranged in a certain definite order, the simplest conceptions standing first, and the rest in order of complexity. This order should be carefully drawn out before reading further. Moreover the definitions give complete and distinct knowledge of the things defined, and thus prevent ambiguity and confusion. In other words they mark out-define—the precise meaning to be attached to the terms employed, and secure that they shall be used in this sense and no other throughout the work. Here then we see the use of the definitions. They state the elementary properties of space which are the basis of geometry, and they give clear and complete knowledge of the things defined.
POSTULATES. In order to assist in deducing new truths from the definitions, Euclid frequently employs certain constructions, and before using them he generally shows how they are to be effected. There are three constructions however which he does not attempt to effect, but simply assumes that they can be done, and demands that he may use them. These are the three postulates, and, having limited himself to these, all other constructions are effected by their aid. Had Euclid thought good to assume other postulates than these, or to employ constructions without stopping to show how they were to be effected, the subsequent demonstrations would have been very much simplified. But his object seems to have been to reduce his assumptions to the narrowest possible limits, and to demand nothing which he was
able to prove. The result of this self-imposed limitation is, that many of the constructions are very complicated even where the object to be attained is very simple; as for example in prop. 3, where five circles have to be described in order to cut off from a greater line a part equal to a less. The gain of such a system, whatever may be the value of it, lies in the mental satisfaction we derive from seeing a man triumphing over difficulties and bringing the greatest possible number of truths in his subject within the circle of strict demonstration.
These postulates may be regarded as fixing the instruments which Euclid allows, and in this view they amount to the following: 1. An ungraduated straight edge of unlimited length. 2. A pair of compasses which open to any extent but close immediately they are taken from the paper.
It will be seen that neither of these instruments can be used to transfer distances, and also that all constructions must consist of straight lines and circles only.
AXIOMS. From the elementary properties of geometrical figures stated in the definitions we may at once proceed to reason about them and deduce general truths. But this is not the case in regard to straight lines,' 'right angles' and parallel lines.' Concerning these Euclid enunciates other properties in the 10th, 11th and 12th axioms, and these with the corresponding definitions form the basis of all our reasonings respecting them. Like the other axioms, although they may be illustrated, they cannot be proved, and are self
evident to every one who understands the meaning of the language in which they are expressed. It will be observed that these last three axioms just referred to, being geometrical truths assumed without proof, form, with the postulates, the whole of Euclid's geometrical demands, and hence are sometimes classed with the postulates; while the first nine axioms are common notions' which are true of all magnitudes whatever. The twelfth axiom may be more simply expressed thus, "Two straight lines which cut one another cannot be both parallel to a third straight line.”
The definitions and axioms together include all the fundamental principles from which the truths constituting the science of geometry are deduced. The way in which this is effected is now to be explained.
PROPOSITIONS. The truths to be proved, and the constructions to be made, set forth in general language, form propositions which are either theorems or problems. The former propose something which is to be proved, the latter something to be done, as for example: "Any two sides of a triangle are together greater than the third side," and "To bisect a given finite straight line, that is, to divide it into two equal parts."
These statements form what is called the general enunciation of the proposition, after which comes the discussion.
First, the general enunciation is repeated and applied to a particular figure. By this means the attention is limited to a special instance selected for consideration, and the discussion is then conducted with reference to this figure. This forms the particular enunciation.
Next follows the construction, directing what lines and circles are to be drawn to aid in the proof of the theorem or the solution of the problem. Lastly comes the demonstration, which shows by a connected course of reasoning that the statement contained in the proposition is true, or that the problem proposed has been solved.
The following, therefore, exhibits the parts of a proposition, and the way in which they will hereafter be distinguished.
First. The general enunciation stands at the head of the proposition, and is always printed in italics.
Then follows the discussion, which may be divided into three parts to be thus distinguished.
1. The particular enunciation.
2. The construction.
3. The demonstration.
The demonstration is effected by combining the elementary truths contained in the definitions and axioms so as to obtain new truths. These results are then used to establish further and more complex truths, and so the process is continued until all the truths which constitute the science of geometry are demonstrated, and their relation to each other exhibited.
In order to this the first principles are combined in a particular way so as to form what is called a syllogism, i.e., an argument stated at length, and in such a form as to show the truth of the reasoning in the most convincing way. Such a syllogism generally takes the following form, which may therefore be regarded as the pattern of all geometrical reasoning.