November 2, 1888.] 



SCIENCE 



207 



between ihe two regions, but the correspondences are certainly ex- 

 traordinary. 



We sliail hope soon to hear that the excellent and practical work 

 •of the New Jersey Survey has been appreciated by the school-boards 

 of that State, and that copies of this new topographic map and of 

 the appropriate local sheet of the State atlas have been placed in 

 all the high schools and academies. Teachers could then carry on 

 the modern reform in geographic teaching beyond its simple first 

 •step, which involves a map of the school-yard and home town, to 

 the more difficult second step, in which correct maps of larger 

 •areas are needed. Until this is generally possible and actual, re- 

 form in geographic teaching will not go far beyond the merest 

 ■elements of the subject. If Professor Cook is as successful in put- 

 ting the Stale maps into practical use as he has been in surpervis- 

 ling their preparation, we shall owe him a double debt. 



W. M. D. 



TISSOT'S THEORY OF THE PROJECTION OF MAPS. 



The question as to what projection to select for a certain map is 

 one of great importance to the cartographer. As is well known, 

 the earth's surface cannot be represented on a plane sheet of paper 

 without distorting the lengths of lines, and without altering the 

 ■size of surface or of angles ; and for this reason it becomes the 

 •duty of the cartographer to select a projection, or a method of rep- 

 resenting the curved surface on a plane, by which the distortions 

 and alterations become as small as possible. Merit is due a French 

 geographer and mathematician, M. A. Tissot, for having first 

 pointed out a inethod by which this problem can be easily solved. 

 Unfortunately his book, which was published in 1881,' is little 

 known, and therefore the necessary process of replacing the old 

 ■projections, which he has proved to be inadequate, by new and 

 better ones, is making hardly any progress. 



The principle which underlies his researches is so clear and 

 simple, that it may be stated here. Tissot assumes an infinitely 

 small circle on a curved surface. If this surface is represented on 

 a plane, the circle assumes the shape of an ellipse, on account of 

 the unavoidable distortion. The great and small diameters of this 

 ellipse are a and b, and their ratio is a measure of the angular dis- 

 tortion, while their product is a measure of the alteration of sur- 

 face. The ratio between a and the radius of the original .small 

 <;ircle, r, is a measure of the alteration of scale. Tissot shows how 

 to compute the length of the axes of this ellipse, which he calls the 

 indicatrix, as indicating the distortion, and how to determine their 

 direction. 



This general theory is next applied to the construction of maps. 

 For any law according to which a system of meridians and paral- 

 lels is constructed, we can compute a and b as functions of latitude 

 and longitude, and thus a means is obtained of studying the dis- 

 tortions all over the surface of the map. 



Maps are made to serve various purposes. In many cases it is 

 necessary that a square inch on one part of the map should repre- 

 sent the same area as a square inch on any other part of the map, 

 •or, as it is generally expressed, that the areas should be preserved. 

 Projections of this kind are called ' equivalent,' while Tissot in- 

 troduces the expression ' authallic' It is evident that every pro- 

 jection in which the indicatrix-ellipse is equal to the small circle, is 

 •equivalent. In other cases it is desirable that each small part of a 

 map be similar to the corresponding part of the earth. This is 

 possible only when the indicatrix is a circle ; that is, when a ^ b. 

 These projections are called by Tissot ' autogonal,' as the angles 

 are preserved. In still other cases we do not mind an alteration of 

 angle and surface, but wish to preserve the length of lines as much 

 as possible. For this purpose the ratio of a, b. and ;" must be as 

 near i as possible. Tissot calls projections in which angles and 

 surfaces are altered ' aphylactic' 



The problem, according to this, is very simple. According to 

 the purpose for which a map is intended, we choose one of the 

 three classes of projections. It is the task of the cartographer to 

 select the projection for a map so that, if one property is preserved, 

 the others are changed as little as possible. If, for instance, the 

 areas are preserved, the angles musi be altered as little as possible. 



> M/moire sur ia Representation des Surfaces et les Projections des Cartes 

 Ciographigues. Par M. A. TlssoT. Paris, Gauthier-Villars. 



A projection which has this property is called by Tissot ' perigonal,' 

 while an autogonal projection in which the alteration of surface 

 is a minimum is called ' perihallic' We have seen that the distor- 

 tion is a function of latitude and longitude. If, then, a country of 

 limited extent is given, we must study this function over the whole 

 area of the map ; and, as there are an infinite number of each class 

 of projections, we are able to select the function so that the un- 

 avoidable distortion of one of the elements becomes a minimum. 



The last case, that of ' aphylactic ' projections, has been treated 

 by Airy in his projection by balance of errors; but the theory of 

 these projections and their application to certain areas has first 

 been given by Tissot. His admirable work inust form the basis of 

 all future cartographic work. 



The importance of his researches may be understood by his dis- 

 cussion of the distortions of the map of France. The great map 

 of the war department of that country is constructed in Bonne's 

 projection ; the map being equivalent, and the maximum alteration 

 of angle being 18 minutes, and the greatest distortion of scale -^X^. 

 These would have been 10' 30" and -^X-g respectively, if a more 

 suitable central meridian had been selected ; but they would have 

 been reduced to 25 seconds and y-jV , if Tissot's principles had 

 been applied. 



It is to be hoped that the thorough study of his work will lead to 

 the adoption of better projections than those which are at present 

 in use. 



SCHOOL-WORK AND EYESIGHT. 



Five Per Cent of Near-sighted Children in an Old, Badly Illuminated 

 and Ventilated School Building, and only 2.8 in a New, Well- 

 arranged Building. — School Life, according to Dr. Tiffany of 

 Kansas City, has Little or Nothing to do in the Development of 

 Ocular Anomalies. 



In the chapter entitled ' Our School Systems,' which is one of 

 the most interesting and suggestive of all those that will accompany 

 the forthcoming annual report of the United Stales commissioner 

 of education, the effect of school-work on eyesight will be very fully 

 discussed, chiefly in extracts from the reports of city school super- 

 intendents. 



Mr. George Howland. superintendent of Chicago schools, says : — 



•' In the old school-rooms, and we need not go far back for 

 them, the light was often so insufficient, that much harm undoubt- 

 edly resulted to the eyes of the children. But in our newer build- 

 ings so much thoughtful attention has been given to this subject, 

 that the evil no longer exists there. Pupils, too, have been allowed 

 to study with too little regard to position, and with the object too 

 near the eye ; perhaps with the result of myopia in some cases, 

 but by no means, in my judgment, to the extent often charged. 



'• The oculist is too definite, and too certain in his knowledge. 

 Why should the book or paper always be ' fifteen inches from the 

 eye ' ? Five feet seven may be the average height of a man, and 

 eight the right number for his boot ; but is he to be considered de- 

 formed, or a monstrosity, who is five feet six. or who wears a num- 

 ber seven or nine ? 



" Of over eighty thousand children in our schools, I have never 

 seen one voluntarily take that distance, and have eminent profes- 

 sional opinion that such an enforced rule would work more harm 

 than ever our neglect has done. Nothing will lie so unblushingly 

 as figures." 



The following, from the report of the board of education, de- 

 scribes the results of a recent examination of the eyes of the pupils 

 of two of the leading public schools of Memphis, Tenn : — 



" The eyes of 681 pupils have been examined. Of these, 588 had 

 perfect sight, Co had imperfect sight from general causes, and 30 

 had impaired vision from eye-strain. It is interesting to trace the 

 gradual increase of this form of impaired sight (near-sightedness) 

 from the primary^ classes, where it is hardly noticeable, to the high- 

 est grade, where it reaches fifteen per cent. In this particular my 

 results are similar to those obtained by examiners in this and other 

 countries. But a point which should not be overlooked is this, — 

 that my examinations were confined to pupils in two different 

 school-buildings, each of which may be taken as a sample of its 

 class. The Market Street building has been recently constructed, 



