Dr, Adamson on Geometrical Demonstrations. 335 



Postulates are therefore inadmissible, except they be of such a 

 character as determines the whole processes of the science ; such 

 as, that things defined or proved may be exemplified, or that ob- 

 jects may be conceived to be superimposed, &c. Any proper- 

 ties becoming the foundation of argument must belong to the 

 nature of the things argued about, and must be found in defini- 

 tions, or in conclusions from them. 



A definition ought to express a single property of the thing 

 defined. To object to a definition that it contains something 

 which may be proved, is futile; for any property may become a 

 conclusion, if some other be assumed as the hypothesis. But a 

 definition must not contain both a hypothesis and a conclusion, 

 as is the case with some proposed definitions of parallehsm, such 

 as, their equidistance at indefinite points, or their common incli- 

 nation to indefinite straight lines. 



Things ought to be considered in the first instance individu- 

 ally, and as they are in themselves, without involving them in 

 combinations. Common sense or sound logic, therefore, requires 

 that when, in constructing definitions, we find more properties 

 than one of an object offered to our choice, we should prefer 

 that which, assumed as a hypothesis, affords the shortest and 

 clearest arguments. Thus there are three properties character- 

 izing straightness of line. Now if we assume as the defining 

 property minimum distance, expressed in the formula ^' shorter 

 than any other between two points,^^;We establish at once, — • 

 1st, the ^vo^Qvtj oi singleness ; 2nd, that of coincidence. We 

 cannot attain the same result, of thus instantly proving all, if 

 we assume either of the others as the definition. 



When the hypothesis of a proposition is single, it can have 

 but one converse statement. Such propositions ought perhaps 

 to be termed corollaries. A normal theorem is a proposition in 

 which the hypothesis is twofold. Its demonstration is the lo- 

 gical combination of these two truths. There are therefore three 

 truths so related, that two being assumed as hypothesis, the other 

 follows as conclusion. Hence such a theorem has two converse 

 forms. Now it is obvious that when the hypothesis announced 

 is, strictly and solely, that which is used in the argument, and 

 the conclusion announced is, strictly and solely, that which is 

 gained by the combination, all converses are necessarily true. 

 All apparent exceptions to this arise from incorrect statements 

 of hypotheses, or of conclusions. This logical position may 

 therefore legitimately be assumed as rendering the demonstra- 

 tion of converses unnecessary. Nevertheless the proof of a con- 

 verse is always to be sought for, as completing the train of ar- 

 gument and determining the real nature of the hypothesis and 

 conclusion, as they are involved in the argument. 



