w 



Oct. lo, 1889] 



NATURE 



573 



Sailing Flight of Large Birds over Land. 



The explanation of this, in Nature (p. 518), seems in- 

 adequate. 



If a large body of air be moving imiformly both in respect of 

 direction and of velocity, no matter at what rate, it might as 

 well be perfectly motionless, in respect of its ability to aid the 

 flight of a bird that is simply floating in it. But in fact the air 

 never is motionless : it moves with the earth, from west to east, 

 at the rate, let us assume, of 500 miles an hour. To a bird 

 floating in the air, whether the earth beneath it moves exactly 

 as the air moves, or not, must be a matter of perfect indiffer- 

 ence. The earth's relative motion does not affect it. I must 

 myself adhere to the explanation which I gave in a former number 

 of Nature (vol. xxviii. p. 28), that the birds avail themselves 

 of differences of the movement of the air, in respect of velocity, 

 or of direction, or of both. Mr. S. H Peal has noticed that 

 " flocks which drift over the hills recover their position on the 

 plains by descending to windward." This is simple enough. 

 The wind is flowing from the plains towards the hills. It rises 

 then as it flows, and has many inequalities in its direction and 

 rate. On entering a gorge a narrow current of air would be 

 thrown upwards with very rapid n scent. Of all the inequalities 

 the birds know how to avail themselves. R. CoURTENAY. 



Tean Vicarage, September 28. 



A Remarkable Meteor. 



On Sunday, September 29, at 7.30 p.m., I observed a very 

 brilliant meteor falling nearly perpendicular a little to the west 

 of north. Its progress towards the earth appeared to be much 

 slower than is u-iually the case with such bodies, the heavens 

 being illuminated for several seconds. The meteor was of a 

 bright sapphire hue ; preceding it were a few drops of bright 

 fiery red, whilst following it came a brilliant trail of light. It 

 seems to have been pretty generally observed throughout 

 Ireland, and letters to the Press from counties Roscommon, 

 Galway, Kilkenny, and Kildare, testify to the interest it has 

 awakened in the country. Richard Clark. 



113 Upper Leeson Street, Dublin, October 7. 



THE METHOD OF QUARTER SQUARES?- 



'T'HE method of quarter squares consists in the use of 

 -*■ the formula 



ab = K« + bf - \{a - bf 



to effect the multiplication of two numbers, a and b. If 

 we are provided with a table giving the values of \n" up 

 to a given value of ft, we may obtam, by the aid of this 

 formula, without performing any multiplication, the pro- 

 duct of any two numbers whose sum does not exceed the 

 limit of the table. 



The method is specially interesting on account of 

 the great simplicity of the formula, by means of which 

 a table of double entry may be replaced by one of single 

 entry. How great a transformation is effected by such a 

 change is evident, if we consider that the largest existing 

 multiplication table of double entry reaches only to 1000 

 X 1000, and forms a closely-printed folio of 900 pages, 

 but that a table of quarter squares of the same extent 

 {i.e. of i«2 up to 11 = 2000) need only occupy 4 octavo pages. 

 The disparity becomes even more conspicuous as the 

 limit of the table is extended, for a table of double entry 

 extending to 10,000 X 10,000, would require nearly 100 

 folio volumes ; and one extending to 100,000 x 100,000, 

 would require nearly 10,000 volumes ; whereas the cor- 

 responding quarter-square tables need only occupy 40 

 and 4bo octavo pages respectively. 



The use of a table of squares in effecting multiplica- 

 tions was recognized as far back as 1690, when Ludolff 

 published his large table of squares, extending to 100,000. 

 In the introduction to the table Ludolff explained how 

 it could be employed in multiplications. In order to 



'.Table of Quarter Squares of all Whole Numbers from i to 200,000 for 

 simplifying Multiplication, Squaring, and Extractitn of the Square Root, 

 and to render the Results of these Operations more certain." Calculated 

 and published by Joseph Blater. (London : Trubner and Co., i888.) 



multiply a and b the table is to be entered with a \- b and 

 a - <^ as arguments, and the difference of the correspond- 

 ing squares divided by 4. If « and b are both even, or both 

 uneven, their sum and difference will both be even 

 numbers, so that i(<^ + ^) and i(rtt - <^) will be integers. 

 In either of these two cases we may therefore enter the 

 table with the semi-sum and semi-difference of the 

 numbers as arguments, the product being the simple 

 difference of the corresponding squares. 



It was not, however, till 1817 that a table of quarter 

 squares {i.e. of \n' for argument ti) was published, for 

 the purpose of facilitating multiplications. If n be un- 

 even, \n' consists of an integer and the fraction ^. This 

 fraction ^ may be ignored in the use of the table, for if 

 either a -\- b or a — b is uneven, the other is so too ; the 

 fraction j therefore occurs in both squares, and disappears 

 from their difference. It may therefore be omitted fron^ 

 the table. 



The table of 181 7, which contained the first practical' 

 application of the method, was published by Antoine 

 Voisin, at Paris, under the title " Tables des Multiplica- 

 tions ; ou, Logarithmes des Nombres entiers depuis i 

 jusqu'k 20,000." It is curious that Voisin should have 

 called a quarter square a logarithm : he called a the 

 root, and \(i' its logarithm. His table extended to 

 20,000, and was thus available for multiplications up ta 

 10,000 X 10,000. On the title-page Voisin described it 

 as effecting multiplications up to 20,000 by 20,000. This 

 statement is justified by the formula 



ab = 2{\a^ + ib-"' - l{a - bf\y 



by which the product was to be obtained if the sum of 

 the numbers exceeded 20,000, the method of quarter 

 squares being then no longer available. It is to be ob- 

 served, however, that this formula requires three entries 

 besides the final duplication. 



Almost simultaneously (1817) a similar table, of the 

 same extent, was published independently by A. P. 

 Burger at Carlsruhe. The method was rediscovered by 

 J. J. Centnerschwer, who published a table of the same 

 extent in 1825 at Berlin. In 1832, J. M. Merpaut pub- 

 lished, at Vannes, a table of quarter squares extend- 

 ing to 40,000. In 1852, Kulik (well known for his 

 large table of squares and cubes to 100,000), who had 

 again rediscovered the method, published a table extend- 

 ing to 30,000. In 1856, Mr. S. L. Laundy published, at 

 London, the largest table of quarter squares which had 

 appeared previous to the publication of the present table. 

 Laundy's table extends to 100,000. It was intended that 

 the multiplications should be effected by means of quarter 

 squares if the sum of the numbers did not exceed 100,000, 

 but other five-figure numbers were to be multiplied by 

 means of Voisin's three-entry formula referred to above. 



It is this change of method that has detracted so greatly 

 from the value of Laundy's fine table. It is evident that 

 the table should have been carried to double its actual 

 extent, i.e. to 200,000, so that any two five-figure 

 numbers could be multiplied together by means of the 

 two- entry formula. The late General Shortrede con- 

 structed such a table, but it was never printed. In the 

 work under notice Mr. Blater carries the table as far as 

 200,000 ; so that, more than sixty years after the publica- 

 tion of the first table effecting the multiplication of two 

 four-figure numbers, the extension to five figures has at 

 last been completed. 



The method of quarter squares has had no opportunity 

 of a fair trial in the absence of a table extending to 

 200.000. Considering the many purposes to which 

 Crelle's tables (which give the product of any two three- 

 figure numbers by a single entry) are continually applied, 

 it is perhaps surprising that no general use should ever 

 have been made of a table which in a very small compass 

 gives, by only two entries, the product of two four-figure 

 numbers. Still it is clear that the full power of the method 



