576 



NATURE 



\Oct. lo, 1889 



very similar to that of a table of triangular numbers, the 

 only difference being that the added numbers i, 2, 3, . . . . 

 are each twice repeated. We may also regard the tabu- 

 lated quarter squares as defined by this rule : The quarter 

 square of n is equal, if 71 be even, to the sum of all the 

 uneven numbers less than n, and, if n be uneven, to the 

 sum of all the even numbers less than n. For evidently 

 the series i + 3 + 5 + . . . . -{■ {m - \) — ti^, and the 

 series 2+4 + 64- • • • • + 2// = «- + «. 



By means of this definition of a quarter square we may 

 exhibit the method of quarter squares diagrammatically 

 as follows. 



Taking as examples the products 8X3 and 7 X 4, we 

 have 



qsq 1 1 - qsq 5 = 8 X 3, 



which may be represented by — 



amd 



qsq II - qsq 3 



which may be represented by— 



7X4, 



The number of points in the extreme left-hand column 

 of the difference of the quarter squares is always equal 

 to the smaller of the numbers to be multiplied. If this 

 number is uneven, there will be one middle line contain- 

 ing a number of points equal to the greater of the two 

 numbers ; the points in excess of this number are to be 

 transferred from the line below the middle one to the 

 line next above it, the excess from two lines below is to 

 be transferred to make up the deficiency two lines above, 

 and so on. If the smaller number is even, as in the second 

 diagram, there are two middle lines differing from each 

 other by two points ; one point from the lower of these 

 lines is to be transferred to the upper, three points from 

 the line below the lower middle line to the line above 

 the upper middle line, and so on. 



It will be noticed that the tabulated quarter squares 

 are, as it were, a species of triangular number in which 

 the succeeding lines of points differ by two, instead of 

 by one. as in ordinary triangular numbers {i.e. viewing 

 the matter arithmetically, the quarter squares are de- 

 rived alternately from the series i + 3 + 5 + • ■ • and 

 2 + 4 + 6 + . . . , and the triangular numbers from the 

 series i + 2 + 3 + . . .). It is the fact of the lines dif- 

 fering by two which enables us in all cases to adjust the 

 points in the difference of two quarter squares so as to 

 form a rectangle in the manner indicated above. 



J. W. L. Glaisher. 



NOTES. 



The American Philosophical Society at Philadelphia is about 

 to celebrate the centennial anniversary of the first occupation of 

 its present hall. The celebration will be held on November 21. 

 The Hon. P>ederick Fraley, the President of the Society, will 

 deliver an address in the afternoon of that day, and in the 

 evening a banquet will be given at the Bellevue. The Society 

 was founded in 1743 *' for promoting useful knowledge." 



On November 16, the Naturwissenschaftlicher Verein of 

 Bremen will celebrate the twenty-fifth anniversary of its founda- 

 tion by an evening gathering and a banquet. 



The International Medical Congress will meet next year in 

 Berlin, from August 4 to 10. Inquiries by intending visitors 

 should be addressed to the General Secretary, Dr. Lassar, Karl 

 Strasse, Berlin. The Congress will be divided into eighteen 

 Sections, and the official languages will be German, English, 

 and French. 



The Congress of the International Geodesic Association was 

 opened in Paris, at the Foreign Office, on October 3. M. 

 Spuiler welcomed the delegates, who represented Austria, Bel- 

 gium, Denmark, Spain, France, Greece, Hamburg, Hesse- 

 Darmstadt, Italy, Holland, Prussia, Roumania, Servia, and 

 Switzerland. M. Faye presided. 



The Ethnographic Congress, which held meetings of its 

 various Sections every day last week in Paris, brought its pro- 

 ceedings to a close on Monday afternoon in one of the large 

 halls of the College of France. It was decided that the Con- 

 gress should hold its next meeting at Bucharest in the autumn of 

 1890. 



The next annual meeting of the Mineralogical Society will 

 be held in the apartments of the Geological Society, Burlington 

 House, London, on Tuesday, November 5, at 8 p.m. 



The Committee of the Sunday Lecture Society have decided 

 that twenty-one lectures shall be given, during the winter, in St. 

 George's Hall, Langham Place, on Sunday afternoons at 4 p.m., 

 as in former years. The first lecture, on ' ' The Origin and Uses 

 of the Colours of Animals," with oxy-hydrogen lantern illustra- 

 tions, will be delivered by Dr. Alfred Russel Wallace on 

 October 20. This will be followed by lectures by Mr. John 

 M. Robertson, Mr. Arthur Nicols, Mr. Chas. Cassal, Dr. 

 Andrew Wilson, Prof. Percy Frankland, and Sir R. S. Ball. 



On Monday afternoon, about a quarter to 2 o'clock, a shock 

 of earthquake was felt in Cornwall. It was accompanied by a 

 loud underground noise like thunder, and was felt distinctly at 

 Doubleborough, Boscastle, and Camelford. The earthquake is 

 said to have been severe enough to shake houses, but no harm 

 was done either to person or property. 



The death is announced from Georgetown, British Guiana, 

 of Mr. E. E. H. Francis, Professor of Chemistry in Queen's 

 College, Georgetown, and Analytical Chemist to the Govern- 

 ment, at the early age of thirty-nine. Mr. Francis entered the 

 service of British Guiana in 1875, having formerly been Professor 

 of Chemistry and official analytical chemist in Trinidad. The 

 two posts now vacant in Georgetown are said to be worth more 

 than ;^700 per annum. Mr. Francis was a member of various 

 learned Societies in this country. 



At the recent meeting of the Iron and Steel Institute, Mr. 

 Frederick Siemens made some remarks during t;he discussion 

 of Sir Lowthian Bell's paper on gaseous fuel, of which we 

 gave an abstract last week. We draw attention to these 

 remarks on account of the important influence Mr. Siemens's 

 experiments and inventions have had on the application of gas 

 for the heating of furnaces. Mr. Samson Fox had stated that 

 he did not propose to use water gas alone for furnace purpos^ 

 but mixed with producer gas or other carburetted gas. 

 Siemens said that a gas would thus be produced similar to tl 

 formed in the Siemens gas producer, which he had applied 

 a considerable time to furnace purposes. He had lately beij 

 experimenting with various gases for furnace use, and had coB 

 to the conclusion to divide gases generally into two clas 

 luminous and non-luminous. The latter, the class to whi 



