546 



NA TURE 



\AprilZ, 1886 



Thus I conceive that muItipUcatioii shou'd be defined from 

 the outset in such a manner as would make it applicable to a 

 fractional as well as to an integral multiplier. If I explain that 

 to multiply 6 by 5 is to repeat 6 five times and find the aggre^fate 

 result, my explanation fails when I am asked to multiply 6 by 

 % ; but if I use De Morgan's definition that " multiplication 

 consists in doing with the multiplicand what is done with the 

 unit to form the multiplier," or an explanation of multiplication 

 cast in this form, I have given an explanation equally simple 

 with the former and applicable also to a fractional multiplier. 



Again, in the very beginning of arithmetic, which is counting, 

 I maintain that much would be gained if from the first the child 

 were taught to count, not one, two, &c., but nought, nothings 

 or uro, one, two, three, &c. ; and then if, later on, ordinal 

 reckoning were made to accord with this, though here unfortu- 

 nately language and usage fails to supply the word wanted, for 

 which, for want of a better, I must coin the form zeroth 

 {noughtth or not/iiiigth being out of the question), thus : zeroth, 

 first, second, &c. Then the transition to counting below zero 

 by negative numbers would follow at once as by a natural de- 

 velopment, when the need for it arose. Thus when it came to 

 the notation of numbers, the place of a digit would properly be 

 reckoned from the units as the zeroth place (not \.\i& first), and 

 would be extended naturally by negative ordinal reckoning 

 downwards, when decimal fractions are introduced. 



This leads me to another illustration, which I am also anxious 

 to introduce as a suggestion on its own merits. Prof. Chrystal 

 has complained that to many students even when beginning co- 

 ordinate geometry the idea of the oriler of a term or an expres- 

 sion is unfamiliar. Now it has occurred to me that this is just 

 the word wanted, to replace the five-syllable word " character- 

 istic," which has been used (or perhaps has not been used just 

 because it is pentasyllable) to express the distance of any digit 

 of a number in order from the unit's digit. Let us speak of the 

 unit's digit as of the order o ; the tens, hundreds, &c., digits of 

 the orders i, 2, &c. ; and the tenths, hundredths, &c. , of the 

 orders - I, -2, &c. ; and add to this that a number is said to 

 be of the same order as that of its highest significant digit, and 

 we have a language not only of the utmost use and importance 

 in decimal arithmetic, but also at once applicable by the most 

 natural extension to an algebraical expression arranged accord- 

 ing to the powers of some letter or letters, while it would 

 enable us conveniently to express in language numbers which 

 transcend our ordinary numerical vocabulary, so that, for in- 

 stance, 53 X 10'- might be read as 53 of the 12th order, and 

 53 X io"i- as 53 of the - I2th order, and so on. 



In conclusion I will only add that, if in this paper I have in 

 any parts expressed myself somewhat dogmatically, I have done 

 so in the hope of challenging discussion, and only claim the 

 acceptance of the views which I have tried to express distinctly, 

 \ibnffly, in the event of discussion resulting in a verdict in their 

 favour. 



ON THE METHOD OF STATING RESULTS OF 

 IVA TER ANAL YSES 



n^HE Chemical Society of Washington is desirous of bringing 

 before chemists and others interested, the report of Com- 

 mittee herewith inclosed ; and as Nature has a wide circula- 

 tion in this country, I am authorised to send a copy of the 

 abstract to your journal, hoping you may find space for it. 



A. C. Peale, M.D., 

 Sec. Chem. Soc. Washington 

 (Office of U.S. Geol. Survey) 

 Washington, D.C., February 25 



The Chemical Society of Washington, at the meeting of 

 November 12, 18S5, appointed a Committee to consider the 

 present state of water analysis, and to present a method of 

 stating analyses, adapted for general use, in order that those 

 hereafter published may be readily compared with each other 

 and with future work. This Committee reported February u, 

 1886, and was authorised to prepare an abstract for publication, 

 in order to call the attention of chemists to the subject. The 

 Society earnestly recommends the adoption of the scheme which 

 is herewith briefly presented. The full text of the report will 

 be published in the next Bulletin of the Society. 



[A is tract) 



Water analyses are usually made to answer one of three 

 questions, viz : — 



(i) Is the water useful medicinally? 



(2) Is it injurious to health ? and 



(3) Is it suitable for manufacturing purposes ? 



Many books relating to water were published during the 

 eighteenth century, but accurate chemical analysis was not at- 

 tempted until about 1820. As the earlier analyses were isolated, 

 rare, and made for special purposes, the form of the statement 

 was of little importance if it were only intelligible. At the 

 present time, however, water-analyses are very numerous. An 

 examination of about a thousand shows some forty-two methods 

 of stating quantitative results, there being sometimes three dif- 

 ferent ratios in the report of one analysis. Such discrepancies 

 render comparisons difficult and laborious. The various 

 methods of statement may be classified under the following 

 general forms : — 



(i) Grains per imperial gallon of 10 lbs. or 70,000 grains. 



(2) Grains per U.S. or wine gallon of 58,372+ grains. 



(3) Decimally, as parts per 100, 1000, 100,000, or 1,000,000. 



(4) As so many grammes or milligrammes per litre. 



The last two would be identical if all waters had the same 

 density ; but as the densities of sea water, mineral waters, &c., 

 are much above that of pure water, it is plain that the third 

 and fourth modes are not comparable. The Committee there- 

 fore unanimously recommends — 



(i) That water-analyses be uniformly reported according to 

 the decimal system, in parts per million or milligrames per 

 kilogramme, with the temperature stated, and that Clark's 

 scale of degrees of hardness and all other systems be aban- 

 doned. 



(2) That all analyses be stated in terms of the radicals found. 



(3) That the constituent radicals be arranged in the order of 

 the usual electro chemical series, the positive radicals first. 



(4) That the combination deemed most probable by the 

 chemist should be stated in symbols, as well as by name. 



The abandonment of Clark's scale has been recommended by 

 Wanklyn and Chapman ; and the recommendation made by the 

 Committee does not involve the disuse of his method, but merely 

 the bringing it into accord with the decimal system — the 

 changing from grains per gallon to milligrammes per kilo- 

 gramme. 



The last conclusion (4) was deemed desirable from the 

 frequent confusion in the statement of the iron salts and of the 

 carbon oxides. 



The Committee is unanimously of the opinion that analyses in 

 the form recommended will prove quite as acceptable to Boards 

 of Health and to the public in general, for whom such analyses 

 are often mnde, as if presented in the mixed and irregular forms 

 commonly adopted. 



The Committee also feels sure that the people in genei'al are 

 better able to form a definite idea of the character of a water 

 from a report stated in parts per 100, parts per 1,000,000, &c., 

 than from one expressed as grains per gallon, the latter being a 

 ratio wholly unfamiliar to any but those in the medical or 

 pharmaceutical professions. 



(Signed) A. C. Peale, M.D. 



Wm. H. Seaman, M D. 

 Chas. H. White, M.D. 



ON SOME POINTS IN THE PHVLOGENY OF 



THE TUNIC ATA 

 T N his monograph on the genus Dclioluin, forming one of the 

 recent parts of the series illustrating the Fauna and Flora 

 of the Bay of Naples,' Dr. B. Uljanin gives a sketch of the 

 phylogeny of the Tunicata, some parts of \\hich cannot, I think, 

 be accepted without considerable modification. Uljanin has 

 evidently regarded the subject from the Doliohim point of 

 view, and, in fact, he only introduces the other groups of 

 Tunicata for the purpose of discussing their relationships to 

 Doliolian. Consequently it is not to be wondered at that he 

 should assign rather too central a position to that genus, and 

 give it too much importance relatively to the other groups. 

 What is of more importance is that his scheme shows a course 

 of evolution which seems not altogether in accordance with 



' "Fauna und Flora des Golfes von Neapel." X. Moii.ygraphie : 

 Doliolum, von Dr. Basilius Uljanin. (Leipzig, 1884.) 



