Mar. 2 1, 1872] 



NA TURE 



403 



been already proved in the seventh proposition. Having 

 proved that from a point zvhich is not the centre tliree 

 equal straight lines cannot be drawn to the circumference 

 of a circle\^\'0^. 7), it was wholly unnecessary to prove 

 that the point from which tliree equal straight lines can 

 be drawn to the circttmfereTice must be the centre of the 

 circle (Prop. 9). 



The two theorems are, in fact, contra-positive forms, one 

 of the other ; the truth of each is implied, when that of 

 the other is asserted, and to demonstrate both geometri- 

 cally is more than superfluous ; it is a mistake, since 

 the true relation between the two is thereby masked. 

 There can be no better proof of this than the fact that 

 the above defect in exposition remained undetected for 

 centuries. Another, though less striking, e-xample of the 

 same kind is presented by the 1 6th and 27th propositions 

 of the first book. Few intelligent boys fail on first read- 

 ing the 27th to note the oddity of giving to two parallel 

 lines a dagger-like shape in order to prove indirectly that 

 "if a straight line falling on two other straight lines make 

 the alternate angles equal to each other, these two straight 

 lines shall be parallel." It is certain, however, that few 

 of them ever discover that the proposition has virtually 

 been proved before, that it is in fact the contra-positive form 

 of the 1 6th, since the latter is obviously susceptible of 

 being thus enunciated : " If two straight lines meet one 

 another, a straight fine falling on them will not make the 

 alternate angles equal." 



The late Prof, de Morgan, to whose keen penetration we 

 owe the detection, not merely of the above defects in 

 Euclid, but of many others, strongly and justly insisted 

 upon the necessity of a more logical study of the elements 

 of geometry. 



I do not advocrte the introduction of more y<;;7«(z/ logic 

 into elementary geometry, but simply the cultivation of a 

 logically severer habit of thought, and the more frequent 

 application of those simple rules of reasoning by means 

 of which tedious reiteration may be so often obviated, 

 and, as a consequence, clearness of insight promoted. As 

 an instance of such a rule I may mention that very useful 

 one according to which " the converse of each of a series 

 of demonstrated theorems is necessarily true if of their 

 several hypotheses, as well as of their predicates, it can 

 be said that one must be true, and that no two of 

 them can be so at the same time." A conviction 

 of the general validity of this rule is readily imparted, 

 even to your pupils, by first selecting familiar instances 

 and then generalising ; and, once imparted, they are put 

 in possession of the instrument whereby converse propo- 

 sitions in geometry are most freciuently and satisfactorily 

 established. 



In conclusion, I may observe that it is chiefly by the aid 

 of general rules, such as those just alluded to, that the 

 mechanical details ;0f demonstration become sufficiently 

 subordinated to allow a complete grasp of the whole sub- 

 ject to be acquired ; they serve, in fact, as the thread on 

 which the isolated propositions of geometry, like beads, 

 have to be strung before they can be properly viewed. 



THE YELLOWSTONE PARK 



THE following, reprinted from the " Reports to Con- 

 gress" of the United States, will serve to show the 

 zeal displayed by the American Government for the im- 

 provement of the people. We regret that we are unable 

 to reproduce the accompanying maps : — 



" The Bill now before Congress has for its object the 

 withdrawal from settlement, occupancy, or sale, under the 

 laws of the United States, a tract of land fifty-five by sixty- 

 five miles, about the sources of the Yellowstone and 

 Missouri Rivers ; and dedicates and sets it apart as a 

 great national park or pleasure-ground for the benefit 



and enjoyment of the people. The entire area com- 

 prised within the limits of the reservation contemplated 

 in this Bill is not susceptible of cultivation with any 

 degree of certainty, and the winters would be too severe 

 for stock-raising. Whenever the altitude of the mountain 

 districts exceed 6,000ft. above tide-water, their setrlement 

 becomes problematical unless there are valuable mines to 

 attract people. The entire area within the limits of the 

 proposed reservation is over 6,000ft. in altitude, and the 

 Yellowstone Lake, which occupies an area 15 miles by 22 

 miles, or 330 square miles, is 7,427ft. The ranges of 

 mountains that hem the valle\ s in on every side rise to the 

 height of 10,000ft. and 12,000ft, and are covered with snow 

 all the year. These mountains are all of volcanic origin, 

 and it is not probable that any mines or minerals of value 

 will ever be found there. During the months of June, 

 July, and .'\ugust,the climate is pure and most invigorating, 

 with scarcely any rain or storms of any kind ; but the 

 thermometer frequently sinks as low as 26°. There is 

 frost every month of the year. This whole region was in 

 comparatively modern geological times the scene of the 

 most wonderful volcanic acdvity of any portion of our 

 country. The hot springs and the geysers represent the 

 last stages — the vents or escape-pipes — of these remark- 

 able volcanic manifestations of the internal forces. All 

 these springs are adorned with decorations more beautiful 

 than human art ever conceived, and which have required 

 thousands of years for the cunning hand of nature to 

 form. Persons arc now waiting for the spring to open to 

 enter in and take possession of these remarkable curi- 

 osities, to make merchandise of these beautiful specimens, 

 to fence in those rare wonders so as to charge visitors a 

 fee, as is now done at Niagara Falls, for the sight of that 

 which ought to be as free as the air or water. 



" In a few years this region will be a place of resort for 

 all classes of people from all portions of the world. The 

 geysers of Iceland, which have been objects of interest 

 for the scientific men and travellers of the entire world, 

 sink into insignificance in comparison with the hot springs 

 of the Yellowstone and Fire- Hole Basins. As a place of 

 resort for invalids it wdl not be excelled by any portion of 

 the world. If this Bill fails to become a law this session, 

 the Vandals who are now waiting to enter into this wonder- 

 land will, in a single season, despoil beyond recovery 

 these remarkable curiosities which have required all the 

 cunning skill of nature thousands of years to prepare. 



" We have already shown that no portion of this tract 

 can ever be made available for agricultural or mining 

 purposes. Even if the altitude and the climate would 

 permit the country to be made available, not over fifty 

 square miles of the entire area could be settled. The 

 valleys are all narrow, hemmed in by high volcanic 

 mountains like gigantic walls. 



" The withdrawal of this tract, therefore, from sale or 

 settlement takes nothing from the value of the public 

 domain, and is no pecuniary loss to the Government, but 

 will be regarded by the entire civilised world as a step of 

 progress and an honour to Congress and the nation. 



Department of the Interior, 



Washington, D. C, January 29, 1872 

 Sir, — I have the honour to acknowledge the receipt of your 

 communication of the 27th instant relative to the Bill now pend- 

 ing in the House of Kepresentatives dedicating that tract of 

 country known as the Yellowstone Valley as a national park. 



I hnnd you herewith the report of Dr. F. V. Hayden, United 

 States geologist, relative to said proposed reservation, and have 

 only to add that I fully concur in his recommendations, and trust 

 that the Bill referred to may speedily become a law. 



Very respectfully, your obedient servant, 



C. Delano, Secretary. 

 Hon. M. H. Dunnell, House of Representatives. 



" The committee therefore recommend the passage of 

 the bill without amendment." 



