AuousT 24, 1883.] 



SCIENCE. 



241 



increased from 9 h. 55 m. 33 8., to 9 h. 65 m. 38 s. 

 The future observer should attend more carefully to 

 what he sees, and tliporize afterward. 



French observations on the solar eclipse of 

 May 6, 1883. 



BY DK. J. .T.VXSSKX OF PARIS, FRANCE. 



A LETTER from the French astronomer Dr. Jan- 

 ssen, who passed through this country on his return 

 from an eclipse expedition, was addressed by him for 

 the use of the as.sociation to Professor Eastman, who 

 translated it, and read the translation in Section A. 

 It was thus entered as one of the papers. Dr. Jan- 

 ssen says, — 



"The princip.il object of the observations was the 

 study of the dark rays in tlie corona. The visibility 

 of these rays depends more on the light-power of the 

 instrnment than upon tlie perfection of the images. 

 At first the ordinary brilliant rays which the corona 

 presents were recognized; but what was new, and 

 more complete than ever expected, was that the back- 

 ground of the coronal spectrum presented the Fraun- 

 hofer's spectrum. All the dark rays were theoretically 

 visible. Phenomena were observed, which indicated 

 that there were some portions of the corona which 

 reflected, much more abundantly than others, the 

 • light emanating from the solar sphere: this would 

 Indicate the existence of cosmic matter circulating 

 around the sim. The rings of Rispighi were not 

 found arranged symmetrically around the sun. The 

 light of the corona was strongly and radially polarized. 

 All these things were associated with the problem of 

 circumsolar cosmic matter. The observations went 

 to show that no important intra-mercurial planet 

 exists." 



Some hitherto undeveloped properties of 

 squares. 



BY O. 8. WESTCOTT OF CHICAGO, ILL. 



The paper began by ascribing due credit to a 

 method for obtaining squares and square roots, de- 

 scribed by Samuel Emerson In ISO."). The principles 

 and details of that method were briefly summarized. 

 Mr. Westcott then stated the general principles of his 

 own method, which is very expeditious. He first 

 shows that ihe tens and units figures of all perfect 

 squares of numbers, from 26 to 49 inclusive, are the 

 same as the tens and units figures of perfect squares 

 of numbers from 24 to 1 inclusive. A table is pre- 

 sented as follows: 



■ (24)'^ = 576, add 100, = 676 = (26)2 

 (23)- = 529, add 200, = 729 = (27)- 

 (22)2 = 484_ add 300, = 784 = (28)- 

 and so on, to 



( 1 )= = 1, add 2400, = 2401 = (49)^ 



To determine the square of any number between 

 25 and .'JO, find the corresponding number below 25, 

 and augment its square by the number of hundreds 

 indicated by its remoteness from 25. Or, more con- 

 veniently, take the excess above 25 as hundreds, and 



augment by the square of what the number lacks of 



.50. 



Thus: (43)- = {Hi - 25) . 100 -I- (50 - 43)= 



= 1S(10 -1- 49 = 1849 



Conversely: To obtain the square root of 1764. 

 The root is plainly between 25 and .50. The tens and 

 units figures indicate 8. Therefore the square root 

 of 1764 is .50 — 8 -42. 



It is further observable, that the tens and units fig- 

 ures of perfect squares of numbers from 51 to 99 in- 

 clusive, are the same as the tens and uniUs figures of 

 the squares of numbers from 49 to 1 inclusive. Since 

 4 X any number of hundreds -I- 25, 50, or 75. gives an 

 ex.iet number of hundreds, it follows that the tens 

 and units figures of the squares of numbers less than 

 25 represent all the possible combinations of figures 

 in those orders of units for nil square numbers. The 

 terminations of all perfect square numbers are 22 in 

 all: viz., 00, 01, 04, 09. 16. 21, 24, 25, 29, 36, 41, 44. 

 49, 56, 61, 64, 69, 7(i, 81', 84, 89, 90. 



The following rule is then deduced: To square any 

 number from .50 to 100, take twice the excess above 

 .50 as hundreds, and augment by the square of what 

 the number lacks of 100. 



Thus : (89)2 = 200 (89 - 50) -)- (100 - 80)= 



= 7800 -I- 121 = 7921 



Conversely, v'3249 : The root is plainly between 50 

 and 60 ; the tens and units figures indicate 7 ; there- 

 fore v'3249 = 50 -1- 7 = 57. 



For greater convenience it is noted, that in such a 

 ease as v'7U21 the root is 50 -f- 39 or 100 — 1 1, and it is 

 easier to use the latter form. That is, if the root 

 is in the fourth quarter of the hundred, subtract the 

 number indicated by the tens and units from 100, and 

 the difference is the root. Thus v'8281 = 100 — 9 = 

 91. 



To square any number from 100 to 200, take four 

 limes the excess above 100 as hundreds, and augment 

 by the square of what the number lacks of 200. 



To square any number from 125 to 250, take one- 

 half the excess above 125 as thousands, and augment 

 by what the number lacks of 250. 



By a series of steps of this character, the author 

 gives methods for squaring higher numbers, and con- 

 versely for obtaining their square roots. A choice of 

 methods is also indicated. The facility which was 

 obtained by such means was deftly illustrated on the 

 bl.iekboard by the author, who in a few seconds per- 

 formed such exploits as raising 5 to the Uilh power, 

 and then showed in detail the processes which ho had 

 mentally executed. The paper sets forth the reason 

 for each rule, deducing it from the usual binomial 

 theorem, with almost obvious simplicity. 



The demonstrations were received by -the section 

 with hearty applause. In response to an inquiry, Mr. 

 Westcott stated, that he had been very successful in 

 te.'iching this method in classes, about a tenth of his 

 pupils becoming rapid experts in the methods of 

 solution, which were especially useful in handling 

 quadratic equations, an<l determining at a glance 

 whether a given number is or is not a perfect square. 



