propositions as are only known to be true in the instances in which experiment has actually been made. For example, a cooper knows that in every instance where he has tried it, the distance that went exactly round the rim of his cask at six times, was the distance to be taken in his compasses in order to describe the head that would fit. But he does not know the reasons why this will necessarily be the case, not only in the instances which he has tried, but in all which he has not tried also. And to supply these reasons, is the object of Euclid's 15th Proposition of the Fourth Book. XXVIII. The conduct of a regularly-ordered demonstration in geometry, divides itself into five parts, which succeed one another, and are named, as follows:— The first conveys a statement of the universal proposition to be finally established. Which is called the enunciation. The second presents to the eye, or to the imagination, a particular instance, in respect of which the truth of the proposition is to be established; with an understanding always, that nothing shall be done in respect of this particular instance, which would not be equally applicable to any other particular instance that could be substituted. Which is called the specification. The third performs, or supposes to be performed, such operations [as, for example, draming or dividing lines, describing figures, &c.] as are to be made use of in the further progress of the demonstration. Which is called the construction. Sometimes the construction, or part of it, precedes the specification. And sometimes no construction is required. The fourth derives from all that has preceded, the establishment of the proposition in the instance presented in the specification. Which is called the proof. The fifth extends the conclusion to all instances that come under the terms conveyed in the enunciation. Which is called the generalization. XXIX. When all the instances to which a proposition may be applied, cannot be included under one specification or one construction, the proposition is said to divide itself into two or more Cases ; which may in fact be considered as so many distinct propositions, each of which has, or is capable of having, its separate enunciation, specification, construction, &c., the which, taken together, amount to the establishment of the universal proposition. XXX. What is called the converse of a proposition is, when the premises and the conclusion are made to change places, and the proposition so arising is presented as a new proposition. For example, if the original proposition is, that magnitudes which are equal to the same, are equal to one another; the converse of this proposition is, that magnitudes which are equal to one another, are equal to the same. XXXI. What is called the negative of a proposition is, when a negation is inserted both in the premises and the conclusion, and the proposition so arising is presented as a new proposition. For example, if the original proposition is, that if of equals one be greater than some thing else, the rest are severally greater than the same ; the negative of this proposition is, that if of equals one be not greater than some thing else, the rest are severally not greater than the same. XXXII. What is called the contrary of a proposition is, when both the premises and the conclusion are altered, not merely by the insertion of a negation, but by being changed into something of a positively contrary kind. For example, if the original proposition is as in the last article ; the contrary of this proposition is, that if of equals one be less than some thing else, the rest are severally less than the same. Scholium. Neither the converse, the negative, nor the contrary of any proposition, is to be admitted to be true, till it has been demonstrated as a distinct proposition. For till this be done, it is impossible to know whether it is true or not. For example, it is shown in the sequel, that if one angle of a triangle is greater than a right angle, the other two are necessarily less than right angles. The converse of which is, that if two angles of a triangle are less than right angles, the other is necessarily greater than a right angle. The negative is, that if one angle of a triangle is not greater than a right angle, the other two are not less than right angles. The contrary is, that if one angle of a triangle is less than a right angle, the other two are greater than right angles. Every one of which is totally and absolutely false. (The Nomenclature of the First Book is continued nithout interruption of numbers in page 50, after the Intercalary Book.) PROPOSITION I. THEOREM.–Magnitudes nhich are equal to the same, are equal to one another. Let A and B be two magnitudes, each of [] which is equal to C. A and B are equal to one another. cD *I. Nomen. For because A i 1* to C, if thei B[T] clature 14. s equal” to C, if their boundaries were applied to one another A and C would coincide ; or else might be made capable of doing so, by a different arrangement of parts. And because B is equal to C, in like manner would B and C. But because A and B would each coincide with C; if the boundaries of both could be applied to those of C at once, A and B would coincide with one another; wherefore A and t!. Som.14, B are equal #. And in the same manner if the magnitudes equal to C were more than two. And by parity of reasoning, the like may be proved in every other instance. Wherefore, universally, magnitudes which are equal to the same, are equal to one another. Which was to be demonstrated. CoRollARY 1. If of equals, one be equal to some thing else, the CoR. 2. If of equals, one be greater, or less, than some thing else; the rest are severally greater, or less, than the same. Or if some thing be greater, or less, than one ; it is greater, or less, than each of the others also. Let A and B be equal, and let B be greater A than C. A shall also be greater than C. B # I.Nom.15. For since B is greater; than C, a certain magnitude may be taken from B, and the remainder be equal to C; or B is equal to the sum of the magnitude equal to C, and of a certain magnitude besides. But because A and B are equal, A (by Cor. 1) is also equal to the sum of the magnitude equal to C, and of the certain magnitude besides. Therefore a certain magnitude may be taken * I.Nom, 15. from A, and the remainder be equal to C.; or A is greater" than C. And in a similar manner if one were less. Or if C is less than B, it is less than A also, For B is greater than C; therefore (as has been proved above) A also is greater than C; that is, C is less than A. And in a similar manner if C were greater than B. CoR. 3. Magnitudes which are equal to equals, are equal to one another. For A is equal to B, and D is equal to B ; therefore (by Prop. I.) CoR. 4. Of magnitudes, if to equals be added the same, the sums are equal. Let A and B be equal magnitudes, to each of which, one after the First Case ; If A and B are such that on their *I. Nom. 14. sum of B and C, would coincide with it, and therefore is equal" to it. Second Case; If either C be added in some other manner than as above; or if A and B would not coincide with one another, but are only such as by a different arrangement of parts might be made to do so; the magnitudes which are the sums, may be made capable of coinciding by merely altering the arrangement of parts, and therefore +I.Nom. 14. they are equalif. CoR. 5. If equals be added to equals, the sums are equal. Let A be equal to B, and C to D. The sum of For because A is equal to B, the sum of A and B D CoR. 6. If unequals be added to equals, the sums are unequal. And that sum is greatest, in which the unequal was greatest. For, of the unequals, one is greatest; that is, it is equal to the other and a certain magnitude besides. But if to the equals were added equals, the sums (by Cor. 5) would be equal; therefore because to one is added a certain magnitude besides, that sum is made greater. So also [with slight verbal alterations] if unequals be added to the same. CoR. 7. If equals be taken from equals, the remainders are equal. For, if this be disputed, let it be assumed that one remainder is greater than the other. Add each to the equals that were before taken away; and because to equals unequals are added, that sum in which the unequal was greatest, (by Cor. 6) must be greater than the other. Which is impossible; for the things by the hypothesis were *I.Nom,26. equal to begin with. The assumption", therefore, which involves this impossible consequence, cannot be true; or one remainder cannot be greater than the other. And because one is not greater than the other, they are equal. So also [with slight verbal alterations] if equals be taken from the same, or the same from equals. CoR. 8. If equals be taken from unequals, the remainders are unequal. And that remainder is greatest, in which the unequal was greatest. For, of the unequals, one is equal to the other and a certain magnitude besides. But if the equals were taken from equals, the remainders (by Cor. 7) would be equal; therefore, because to one of the objects from which subtraction is made, is added a certain magnitude besides, that remainder is made greater. So also [with slight verbal alterations] if from unequals be taken the sa/226. CoR. 9. If unequals be taken from equals, the remainders are unequal. And that remainder is least, in which the unequal was greatest. For if, instead of the unequals, magnitudes equal to the smallest of them were taken from the equals, the remainders (by Cor. 7) would be equal. But because from one is taken a certain magnitude besides, that remainder is made less. So also [with slight verbal alterations] if unequals be taken from the sole. CoR. 10. Magnitudes which are double of the same or of equal magnitudes, are equal to one another. And so if, instead of the double, they are the treble, quadruple, or any other equi multiples. Let A and B be equal magnitudes. The double A C of A is equal to the double of B. B D — For, to take the double of A, is to add to it a magnitude C that is equal to A ; and to take the double of B, is to add to it a magnitude D that is equal to B. But because A and B are equal to one another, (by Cor. 3) C and D are equal to one another. Therefore (by Cor. 5) the sum of A and C is equal to the sum of B and D. And in like manner if to the sum of A and C be added another magnitude equal to A, and to the sum of B and D another magnitude equal to B. And so on. |