i8 



NATURE 



[May 4, 1893 



made by a citizen of London. Recorde dedicates the first English 

 .nlgebra to the company of Merchant Adventurers trading to 

 Muscovia. 



Important advances in mathematics vfere made by the pro- 

 fessors at the college in London, founded by Sir Thomas 

 Gresham. This feeling among the trading classes produced 

 results in Italy which Libri tells us were unparalleled in any 

 previous time. We all know of the Floral Games of Toulouse, 

 and the athletic contests of the Greeks at Olympia and Corinth. 

 But Libri tells us that just this interest, just this popular ex- 

 citement was felt in Italy when Ferrari or Bombelli had made 

 a step in advance in the solution of cubic and biquadratic equa- 

 tions. There were public challenges to contests of skill, pro- 

 clamations by heralds, wagers to be decided. There is a 

 collection of answers given by Tartaglia to questions submitted 

 to him for sohition by men from all ranks in society, princes, 

 monks, doctors, ambassadors, professors, architects, and mer- 

 chants, and a large proportion of them had to do with cubic 

 and biquadratic equations. It may seem rather strange that 

 this particular portion of Algebra should have excited so much 

 interest, but it must be remembered that it is not possible to 

 determine beforehand what researches into abstract truth will 

 afterwards lead to the greatest practical benefits. There was a 

 widespread belief that the new powers of calculation would 

 bring about material advantage. 



I trust that I may be pardoned for thus bringing forward matters 

 which are no doubt very familiar to most of the members of 

 this Association ; but the object has been to give a sample of 

 the kind of facts that would be likely to appeal to the minds of 

 young learners, and to attach some human interest to the ab- 

 stract subjects they are studying. This human int,erest is to be 

 found in the history of navigation not less than in that of com- 

 merce. The relation between the commercial impulse and the 

 navigation impulse was not exactly one of succession. The 

 former was the earlier, then the two for a time went on together, 

 and afterwards the latter was supreme as a ruling motive for 

 promoting mathematics. 



The two great problems in'navigation were first, if you knew 

 where you were, to find how you could best get somewhere 

 else ; and secondly, if you did not know where you were, to find 

 this out by astronomical observation. The solution of the first 

 was mainly dependent on maps and charts, and consequently 

 for a long time men were hard at work making these for the use 

 of sailors. The first great promoter of this work in modern 

 times was Prince Henry of Portugal, called the Navigator, and 

 after his death in 1460 to the close of the century, Portugal, 

 eagerly engaged in the exploration of the coast of Africa, con- 

 tinued to be the great chart-producing country. Later on it was 

 to the Netherlands that we were principally indebted for im- 

 provements in this direction, and in the long list of those thus 

 engaged a prominent place is taken by Stevin. Mercator's 

 projection is so called from Kauffman, who invented it in 1566, 

 but did not clearly show the principles on which it is founded, a 

 task that was afterwards accomplished by an Englishman, 

 Edward Wright, whose great services to science have been but 

 scantily recognised. 



The second great problem — to find out where you are by 

 astronomical observation — was a pressing question in the six- 

 teenth and seventeenth centuries. The chief instrument the 

 Elizabethan mariner had at his command was the astrolabe. 

 This was- made in very various forms. For use at sea, of course 

 the simplest form was chosen. There is a plate in Mutton's 

 Mathematical Dictionary of one, consisting of a graduated circle 

 held up by a ring, and so keeping a vertical position by its own 

 weight, furnished witli an arm and two sights, by which the 

 altitude of the sun, moon, or stars could be estimated. The 

 astrolabes in use on land were fitted up with much greater 

 refinement. 



An instrument perhaps more frequently used, easier to work 

 with than the astrolabe, but less accurate, was called the cross- 

 staff or fore-staff. It was composed of a graduated wooden rod, 

 about three feet long, with cross pieces sliding along it of differ- 

 ent heights, and the angle was observed in the same way that a 

 volunteer uses the sights on his rifle. This fore-stafif could be 

 applied to roughly determine the distance between two stars. 



To determine wi,h any accuracy a ship's place at sea, three 

 things are requisite. First, a theory that is true and workable 

 as far as it goes ; secondly, means of observation ; thirdly, 

 means of calculation. A defect in any one of these requisites 

 renders comparative excellence in the other two o small use. 



NO. 1227, VOL. 48] ■ 



Now, the mariners of Drake's time had scanty theoretical 

 knowledge, poor instruments, and very deficient means of cal- 

 culation. Tney could, in a rough fashion, find out in about 

 what latitude they were ; the longitude remained a mystery. 



It was at ihe beginning of the seventeenth century that the first 

 great improvement took place. The invention of logarithms, 

 by Napier, placed the calculating power at one bound lar in ad- 

 vance of either the theoretical knowledge or the means of 

 observation. His system, further developed by Briggs, the 

 Gresham professor, so completely supplied the want previously 

 existing, that any improvements made between then and the 

 present time are mere matters of detail. 



The improvements in theory and in instruments went on 

 gradually and together. Tycho Brahe did much to advance 

 the efficiency of instruments, and every step in this direction 

 gave the means of correcting or developing previous defective 

 theory, and each theoretical advance suggested or rendered 

 possible some new instrument of observation. It is no proper 

 part of my subject to trace the steps of this progress. It is suffi- 

 cient to say that now the shipmaster, often a man of no great 

 scientific attainments, generally accustomed to work by rules, 

 the reasons for which he does not know, has in his cabin a 

 chronometer and a book of navigation tables, which represent 

 in a material form the genius and the toil of the master minds 

 that have arisen during the centuries of the past. 



In the application of pure mathematics to navigation, as well 

 as to many other purposes, it is curious to notice the changes in 

 the relations between graphic methods and calculation methods. 

 At first the former greatly predominated. The quantit.es of 

 straight lines and curves engraved on Drake's astrolabe, the 

 profusion of scales on old sun dials, that but few thoroughly 

 understand, were originally intended and were accepted as the 

 most simple means of determining practical problems. They 

 gradually gave place to numerical calculation, but not very 

 quickly. Fifty years ago a boy's training in the elements of 

 navigation was conducted far more on the lines of geometrical 

 construction than it is at present. In quite recent times there 

 has been a revival of graphic methods in a somewhat different 

 aspect. Besides the value they have always had for illustration 

 and explanation, it has been seen that there is a special field for 

 them in cases where calculation would be long and troublesome, 

 and this special field is being clearly marked off. 



The correspondence between the practical aims of men and 

 the progress of theoretical knowledge and of means of calcula- 

 tion does not stop with navigation. In recent times the need 

 for more powerful or more exact machinery, the employment of 

 steam and electricity, our increased knowledge of what is meant 

 by heat and light have had the effect of demanding fresh ad- 

 vances in mathematical methods ; or, perhaps, more exactly of 

 selecting from the mass of abstract truth acquired for its own 

 sake the particular portion suited to the special purpose. These 

 influences have had, however, nothing to do with the school- 

 boy's elementary programme, and are, therefore, outside the 

 immediate subject of this paper. 



In conclusion, I would urge that if there is any sound founda- 

 tion for the view s that have been expressed, we ought not in 

 England (o be without some elementary primer of the History 

 of Mathematics. 



FOGS AND HORTICULTURE. 



pROF F. W. OLIVER'S second report on the effects of 

 -^ urban fog upon cultivated plants has been presented to the 

 scientific committee of the Royal Horlicultural Society, and is 

 now printed in the Society's Journal. The following is the 

 passage in wliicli he deals with possible remedial measures : — 



There is very little of what I can say likely to be consoling to 

 the horticulturist. We must recollect that in the employment 

 of measures directed towards mitigating the injuries incident to 

 fog, two factors — the presence of poisons in the atmosphere and 

 the reduction of light — have to be considered. To counteract 

 these the urban cultivator is asked to construct air-tight houses, 

 with definite openings where the admitied air can be filtered; 

 whilst to compensate for the loss of light due to the absorption 

 which the rays undergo in traversing a stratum of dense fog, he 

 must provide a generous installation of electric light. Without 

 doubt, the entire preservation of vegetation in foggy weather is 

 only a matter of £ s. d. But it is for the cultivator to sit down 



