July 8, 1892,] 



SCIENCE. 



Many of the questions which present themselves to our 

 jurists and juries are simply questions of fact, and the testi- 

 mony on which the determination of such questions depends 

 often comes from persons who are neither interested nor dis- 

 honest. In such cases it ought to be easy to reach a true 

 conclusion, but there is often failure, growing out of honest 

 differences of opinion. 



An eminent attorney not long since referred in conversa- 

 tion to a certain decision of the Supreme Court of the United 

 States concerning which there had been a strong dissenting 

 minority. The question was one which involved neither 

 passion nor politics, and he declared that to him it seemed 

 utterly impossible for a disciplined mind to reach other than 

 one conclusion regarding it 



In any review of this subject, such as is here suggested, it is 

 neither necessary nor proper to refer to the numerous in- 

 stances of utter failure in our judicial system, attributable to 

 a lack of integrity on the part of those who administer the 

 laws or to the mischievous results of appeals to passion or 

 prejudice by unprincipled advocates. It is sufficient to recog- 

 nize the fact that failure in the administration of law is not 

 uncommon where witnesses are honest, juries intelligent and 

 well-meaning, and judges incorruptible. 



The rapidly increasing number of controversies within the 

 church, to say nothing of those in which the disputants are 

 on opposite sides of the wall, show conclusively that the 

 logic of the theologian must sometimes go at a limping gait. 

 In political or social economy there is great diversity of 

 opinion among good and able men. Certain financial legis- 

 lation by Congress is honestly thought by many people to be 

 necessary to prevent widespread disaster and the financial 

 ruin of one of the largest and most important classes of our 

 citizens; by other equally intelligent and equally honest men 

 such action on the part of the National Legislature is con- 

 demned as dishonest in principle and sure to be fatal to the 

 business interests of the country. 



A large number of able and patriotic men addre.ss them- 

 selves to the solution of the problem of the adjustment of 

 duties upon imported merchandise. All have access to the 

 same store of experience; the discussions and investigations 

 of the past are open to all alike. In the end, however, their 

 conclusions, even as to elementary principles, are diamet- 

 rically opposed to each other. 



But I have neither the time nor the disposition to enter 

 into an exhaustive examination into the miscarriage of logic 

 in the regions of politics, religion, or social science. I must 

 restrict myself to some consideration of the uncertainty of 

 conclusions reached by what may be broadly included under 

 the general term "the exact sciences," a division of the sub- 

 ject not unlikely, I hope, to be of some interest to membeis 

 of this society. 



At the threshold of the investigation we are confronted by 

 the term " exact sciences,'' and it is of the utmost importance 

 to reach a clear understanding of the meaning of this phrase, 

 in the beginning. By some writers its application is limited 

 to the mathematical sciences or substantially to pure math- 

 ematics. This does not seem, however, to be in accord with 

 the general usage among scientific men, and a wider signifi- 

 cance will be here given to it. 



Pure mathematics may, and possibly must, be regarded 

 as a mode of thought; as symbolic logic; as an abridgment 

 of mental processes by the selection of that which is com- 

 mon to all, and its formal expression by means of signs and 

 symbols. Intellectual operations which, on account of their 

 complexity and length, would be possible only to a few of 



the highest capacity are by the aid of mathematics brought 

 within the range of the many. In virtue of the simple and 

 beautiful nomenclature of the science, one can see at a glance, 

 in a formula or equation, the various relations, primary and 

 secondary, direct and implied, which exist among the sev- 

 eral magnitudes involved, which, if expressed or defined in 

 ordinary language, would be beyond the understanding of 

 most intelligent people. 



The principles and rules governing mathematical opera- 

 tions have been, in the main, so well worked out and so uni- 

 versally agreed upon that in mathematics one can hardly 

 go astray, at least not without the certainty of almost imme- 

 diate detection and conviction at the hands of many skilled 

 in the use of this wonderful intellectual device. When deal- 

 ing with quantity in the abstract, or with matter under just 

 such restrictions or possessed of just such properties as are 

 prescribed, mathematics becomes a machine of certain per- 

 formance, the output of which can only be in error through 

 the conscious or unconscious mistakes of the operator. As 

 such it challenges the admiration of all, and it must forever 

 be regarded as among the first, if not, indeed, the very first, 

 of the few really splendid creations of the human inteUeet. 

 When Plato, in reply to a question as to the occupation of 

 the Deity, answered, "He geometrizes continually," he em- 

 phasized the dignity and the incontrovertibility of mathe- 

 matical reasoning. 



It is no reflection, then, upon the importance and value of 

 the science of mathematics to leave it upon the pedestal 

 which it rightfully occupies, considering it as separate and 

 apart from other sciences. In their development it may 

 and does play a most important part, in which, however, it 

 is identified rather with the investigator than with the sub- 

 ject investigated; for, in studying the elementary principles 

 of abstract dynamics, one may follow the now somewhai 

 antiquated and cumbersome processes of Newton or the more 

 simple and elegant methods of Clifford or Maxwell, but the 

 results will in all cases be the same. 



Before finally dismissing the pure mathematics, however, 

 especial attention must be invited to one or two principles 

 involved in their application by way of contrast with the 

 condition of things which exists in the domain of the other 

 sciences. It is sometimes declared by way of a criticism of 

 mathematics that " what comes out of it is never better than 

 what goes in." In a certain narrow sense this is true, but 

 in a broader and truer sense it is as false as it would be to 

 say that grain and fruit are no better than the soil from 

 which they spring. 



The mathematician has the great advantage over the 

 physicist, the chemist, or the geologist that he not only can, 

 but almost necessarily must, completely define the elements 

 with which he has to deal. If he deals with matter, before 

 he can put it into his equations he must needs restrict it as 

 to form and dimensions and endow it with definite physical 

 properties, the relations of which are capable of analytical 

 expression. If, after this, his power of analysis is sufficiently 

 great, the conclusions which he reaches can have no element 

 of uncertainty in them, provided always they are considered 

 as referring only to the supposititious material with which 

 the investigation was begun. That the conclusions are not 

 in harmony with known phenomena is evidence only of the 

 fact that the material of nature is not the material which is 

 symbolized in the formula, and that certain properties which 

 are common to both are modified in the former by the pres- 

 ence of others which are not attributed to the latter. When 

 MacCullagh, Neuman, Stokes, Sir William Thomson, or Max- 



