UC-NRLF B 4 ESD MAT G) 'b^m Hidiii c,}it Professor of Geography University of California TTrjlil|lll|lll|in|ni|IM|lll|III]TTTpnprTpTppi!|lll|iri|lll|lll|lTT]TlT[TT Tmiiiiiii|niiiii|iii|iii|ii !|lll|lll|lll|lll|lll|lll|lll|lll[lll|lll|lll|lll|lll|lll|lll[Trm]T|TTT|TTT]T^ J ^ 9 01 a 6 ^^ 9 Z 9 9 t 2:i4:S678;^9^^]S' iiiliiiliiilm iiilhilihlniiihl,nhi;liiil 9 ^ 19 8 g f 10 11 12 13 14 15 6 8 n> 9 ^. '19 8 Z> f 10 11 12 13 14 15 16 7 i> 9 1^- 19 8 S T 10 11 12 13 14 15 16 17 8 9 ^ 19 8 Z T 10 11 12 13 14 15 16 17 18 n> ^^ to 8 g f 10 11 12 13 14 15 16 17 18 Id 9 19 8 g f 10 11 12 13 14 15 16 17 18 Id 19 % g g T 10 11 12 13 14 15 16 17 18 Id 19 1^ 19 ^ T 10 11 12 13 14 15 16 17 18 Id 19 1^ 119 8 T 10 11 12 13 14 15 16 17 18 Id 19 1^ It 18 g 10 11 12 13 14 15 16 17 18 Id 19 1^ 1(9 18 IS f 11 12 13 14 15 16 17 18 16 19 1^ 119 18 12 IT 10 '12 13 !l4 15 16 17 18 Id 19 1^^ Vt 18 1% IT 20 TABLE V. Midtiplication and Division. Tonal Si/stem. 1 2 3 4 5 6 7 8 d 9 % 10 8 2 T 10 2 4 6 8 9 19 S 10 12 14 16 18 19 It^ 12 20 3 6 d '19 T 12 15 18 1^- 12 21 24 27 29 28 30 4 8 '(9 10 14 18 110 20 24 28 2i^ 30 34 38 319 40 5 9 T 14 Id IS 23 28 28 32 37 310 41 46 A^ 50 6 10 12 18 U 24 29 30 36 31^ 42 48 42 54 59 60 7 Zo 15 I'LO 23 29 31 38 3T 46 48 54 5^. 62 6d 70 8 10 18 20 28 30 38 40 48 50 58 60 68 70 78 80 d 12 1^ 24 28 36 3T 48 51 59 63 m 75 72 87 do 9 14 IZ 28 32 319 46 50 59 64 62 78 82 8'(9 d6 90 % 16 21 210 37 42 48 58 63 62 7d 48 8T d9 95 ^0 t 18 24 30 319 48 54 60 610 78 84 do 319 98 ^4 190 8 19 27 34 41 U 5^ 68 75 82 8f dio 99 %Q 1^3 80 S 11^ 29 38 46 54 62 70 7S 81^ d9 98 ^6 m 82 20 T IZc 23 319 A% 59 6d 78 87 d6 95 B4 m 82 21 TO 10 20 30 40 50 60 70 80 do 90 ^0 too 80' 20 TO 100 23 EXPLANATION OF TABLES. Table I. shows the different notation of equal numbers in the decimal and tonal systems, where it will be seen that the new system require a less number of figures in expressing a high number; decimal 134zi: 86 tonal, yet the real value is the same in both cases. Table IL is a further extension of Table I. useful for transferring numbers from one system to the other. Example 1. Required how the number 31,868 will be noted by the tonal system ? ■ { 30,000 — 7550 1 1,000 = 3S8 800 — 320 68 =r 44 Decimal, < > Tonal. 3,1868 = 7M(9 Example 2. The year 1859 expressed by the tonal system, will be 739, or it would apparently carry us back over 11 centuries. Example 3. A lady of 35 years, required how old she will be by the tonal system? The answer is 23 years. Table III. is an excellent illustration of the utility of the tonal system. It contains the ordinary fractions used in the shop and the market. It will be seen that the vulgar fractions in daily use, require four to seven decimals, where the tonal system require only one or two figures. It must be admitted that it is more natural to divide things into halves, quarters, eighths or sixteenths, than into fifths or tenths, and when the 24 natural fractions are expressed by decimals, they be- come too complicated for the ordinary uneducated 3 mind, as — is equal to 0.1875, which I am assured, cannot be conceived by the very best arithmeticians, but they know by practice in calculation that it is so. 3 In the tonal system it is very easy to conceive that — is equal to 0.3 Table IV. is for addition and subtraction, arranged in the ordinary way, that where the vertical and hori- zontal columns cross one another is the sum of the index numbers. Example 4. 3 + 5 = 8, 5 + 9 = T, and t-\-Z>^l6. For subtraction, find the greatest number in the column in which the smaller number is the index, and the index of the cross column is the difference, as 17 — t^ = ^^. ADDITION. Ex. 5. < To 36^'9S Add 10-:)78 I I^Same 47526 2 + 8 = 16"] 9 + 7 = 11. ^ + ^ = 14. . 6 + = 6. . . J^ 3+ 1=4 47526 f 89T^ Ex. 6. \ 308 83210 ^ + ^ + 8 = 119 ^ f + g + = lg I 9 + 5 + 3 = 12 4 + 8 = 19 \ 8321'^ J 25 Ex. 7. ' 3819 T3 \ •ST •01 •34 •03 •49 r 678.10 1(1 Ex. 8. 3T(98^61 945 9f 74012 Ex. 1. SUBTRACTION ' From 38S9T ^ Subt. 4^.13 ! UifF. 3431 C Ex. 9. r+ 81042^ 1 — 4210f Ex. t. C+ 89^g0-01f I — 2;00f-301 im\m [ 7(9^(90-812 In all arithmetical operations, the tonal fractions work precisely the same as decimal fractions. Table V is an ordinary arranged mnltiplication table. MULTIPLICATION. Ex. 10. C 389^.6 I 6 154044 80-1T9 72 6 X 6 = 24 6 X ^ = 42 . 6X9 = 3(9. . 6 X 8 = 30 . . . 6X3 = 12 154044 38^206-4f 000684 Ex. 8. \ 1013T4 384586 Ex. Zo. { 3^47154 22f81d3'(9 1105TO3278 . 154742589 . . 171^621191(9 26 DIVISION Ex. f. 3 189ia^^0 83-5.-U9-99 18 182 Ex. 10. 41989-1)008 2901965- 13393 394 09 D 1249 .... 1234 119 . . .. n.. . 16:30 . . 1588 . . n. . ^80. 95(9. 05. to. c148 ^19 20 230 182 20 IS 550 576 20 690 576 1290 1234 fi?on 27 Table of Tonal Logarithms. Number. Logarithm. Number. Logarithm. 1 00 D 'd-m 2 4 9 0-84 3 66 % 0-88 4 8 l^ o-2;6 5 ^5 n 0-^8 6 96 % 0-T4 7 14: T 0-T^. 8 0-'l9 10 100 This table of ional logarithms is a good illustration of the simplicity of the system. In logarithms for single figures, the montissa contains only one or two fonals, where the decimal system has a tail of an endless number of decimals. TONAL SYSTEM OF WEIGHTS, MEASURES, AND COINS. A unit for the measurement of length ought to be of a convenient size for the artizan in laying out work. Units of about the length of the English foot seems to be almost universally adopted, but it will be observed that the artizan generally employs two such units, or a two foot rule, which length appears to be the best suitable for the actual operation of measurement. In some countries tniits of about this length are employ- ed, as the Swedish alu, Russian archin, the elle of Germany, and others, most of them approaching the length of about 2 feet or the footstep of a man. The French meter is the longest unit employed for ordinary measurement in the shop and the market. In accord- ance with my own observation on the actual perform- ance of laying out work with the French meter, and '2S also from comments made by Frenchmen, I believe that the meter is too lour/, to be convenient for the artizan, and when made for the pocket, a great many joints are required, which are objectionable in its application. The meter being divided into 100 parts makes it inconvenient to divide the joints, ten are too many, — four will contain the odd number 25 centi- meters or 2| decimeter in each part, — five parts are not practical. When the artizan applies the meter, he cannot well see the correctness at both ends without placing himself in an inconvenient position, by which the correctness of the measurement is liable to error, — having myself in Paris been witness to the fact alluded to. The French have divided the quadrant of the earth into fixed number 10,000,000 parts in preference to giving the artizan a convenient unit for his measure- ment. The division of the quadrant of the earth is merely once a matter of calculation, and could easily be divided into an odd number, rather than to give the artizan a unit which does not suit him. If the 10,000,000 parts had some even relation with the general division of the earth's great circle, as to the length of one degree or minute, it would have furnished a good reason for the length of the meter. The quad- rant of the earth divided into the most natural or binary divisions hal/s and halfs, would lately arrive to a length of about 23| inches, which would have been a much more suitable unit than the meter which is nearly 40 inches. When a new unit of length is to be selected, it ought to be so adjusted as to bear an even relation to the length of minutes and seconds on the great circle of the earth. By the tonal systefn it would become the most ' tL-'^'^y^f.'^ ' •'^ V \r'WVrsf^ -^ '— ■— ---i^ - , ' ^ -.. . . ' -w^ ^VTX^ ^ \f^ W^ A-Zy-t-^ t^ r' // - '- - y>v t.^- A. /«>*"'/ A ''- /-M.^29 natural to divide the circle of the earth repeatedly by the tonal base. The mean circumference of the earth is about 2J:851-64 miles, or 131216659-2 feet, which latter divided by 16 X 16 X 16 X 16 X 16 X 16 X 16 = 16" (1000,0000 tonal) would be 3,489,767,296 parts, each of a length of 0-48882 feet, or 5-86584 inches. Sup- pose this to be adopted as a unit for the measurement of length, and to be divided and multiplied by the tonal base, its full size appearance will be as shown by fig. 1. (This figure is drawn on the rule fig. 3). Meter seems to be a good and proper name for the unit of length, and will therefore retain that word by calling the new unit the tonal mete?'. It seems to me that the most correct way of ascer- taining the size of our globe, would be to measure the longest straight distance on the land, which by examining the map, we will find is in about 31° north latitude ; starting from Changhae China, through Tchintou, over the Himalaya mountains, Bassora, Isthmus of Suez, Cairo, Cadames, to Santa Cruz, a distance of about 7700 miles, or over 130 degrees in longitude. This distance should be ascertained by actual measurement, compared with astronomical ob- servations, and fixed points located at every tmton. The only obstructions, though not serious, in this distance, are, lake Tai-Hou in China, about 40 miles, and the Himalaya mountains. The same could be repeated in America, from Washington to San Francisco, in the 38th parallel, a distance of about 2240 miles, or about 45 degrees diff'erence in longitude. AVhen these distances are known with their corresponding latitude and longitude, the great mean circumference of our globe is easily 30 calculated by well known rules in mathematics. Every Nation could by the same rule find out the length of the standard tonal meter in their own country. Length. Tonal System. Old System. ' m. =1 Metermill — 0'001432 inches Ens. lOUO -1- m. = 1 Metersan = 0-022913 luu 1 m. = 1 Meterton = 0-366615 10 One Meter =: 5-86584 in.=:0-48882 ft. 10 meter =z 1 Tonmeter z= 7-82112 feet. 100 meter = 1 Sanmeter =125-135 " 1000 meter — 1 Millmeter — 2002-207 It will be perceived that when the word meter is placed before the expression of value, it impresses on the mind a fraction, as meter-mill zn ^^- or r-rr of a mill lUUO meter; and when the expression of value is placed before the unit, it denotes a multiplication of the same, as Tonmeter rz 10 meter. The minute measurements, as wire and needle gauges, to be tonally numbered and divided. In the present Birmina'ham wire o-auo;e the hio^hest number denotes the smallest dimension, which ought to be the reverse. Suppose the meterton to be divided into 100 tonal parts (256 decimal) or metermills, each would be about i of No. 36 B. W. gauge, or 0-001432 of an inch; this part to be noted No. 1 and the meterton would be No. 100 which is about f of an inch. By such arrangement, the very expression of the number impresses the mind of the real size of the minute measure, derived from the main standard, the circumference of the earth 31 Such a gauge would most likely be generally adopted for minute measurements in shops where the present B. W. gauge was never known. The tonal meter to be employed in manufactories, for measuring machinery, &c., corresponding to the English foot. The artizan generally carries a two foot rule, folded into two or four parts, the tonal measure would be of precisely the same shape, but with four units instead of two. On the accompanying plate are full size drawings of the tonal measure, of which fig. 2 is a four-folded rule of one meter in each part, in appearance very much like ordinary four-folded two foot rule. The side A A con- tains the meter tonally divided and numbered. The other side B B of the same rule, contains divisions for circumference and areas of circles, arranged so that opposite the diameter on A is the circumference on B and area on C. Suppose the diameter to be I'tQ meters on A, it corresponds with 5-48 meters the circumfer- ence on B, and 2-5 square meters the area on C. The small divisions between B and C are each four meter- sans drawn from A to assist the transference and read- ing on B and C. Fig. 3 represents a two-folded tonal measure, similar to the English two-folded two foot sliding rule, numbered and divided same as fig. 2. The part E on which fig. 1 is drawn, to receive numbers of specific gravity of substances, and other co-efiicients of general nse in practice. The scales F, G, and H, are the ordi- nary sliding rule, divided into the tonal system, which in this case stands in such relation to the divisions on the side D D, that any number on H, corresponds with its logarithm on D. The operation on the tonal slide 32 rule will be the same as that on the ordinary decimal one.* The clear and simple relation between numbers aiid logarithm in the tonal system has led me to some valua- ble conclusions in reference to calculating machines, and mathematical instruments, which I believe would be of the greatest service to the world. The Tonmeier, (7-82112 feet) to correspond with the Fathom, to be used for measuring ropes, cables, depths of water, &c., &c. The Smimeter (125-135 feet) to be the length of the surveying chain, to consist of 100 (256 decimal) links of one 7netcr each. The Millmeter (2002-207 feet) for road measure and distances at sea, to correspond with miles. One mill- meter is equal to one timmiU., see division of time. Longer distances on the earth's surface would be expressed in Tims. Astronomical distances would be best to express in great circles of the globe, by which the mean distance to the sun would be ^fl circles. Great distances, such as to fixed stars, could be easier conceived by this measure. Time and the Circle. Tonal System. Old System. One circle =10 Tims — 24 h'rs or 360 degrees. 1 Tim - 10 timtons = IJ " 22°30' Itimton zulOtimsans - 5" 37i'- l°24'22i" 1 timsan = 10 timmills = 21-1=- 5' 9" 1 timmill - 1 Millmeter - 1-31836^'^'=°'^'^^- 19-77" The length of a pendulum vibrating timmills will be %'666 meters = 67-975 inches. * A few tonal measures are now being made in Philadelpliia. 33 The time, circle, and compass would thus be equally divided, and greatly simplify all astronomical and nau- tical tables and calculations. In expressing time, angle of a circle, or points on the compass, the unit tim should be noted as integer, and parts thereof as ional fractions^ as 5*86 Urns is live times and mcionhy. The unit Urn to be pro- nounced as in the English word t'miber. The accompanying figures are drawings of a clock or watch dial, and a compass on the tonal sysiem. Fig. 4 shows the appearance of a ional clock dial, the time indicated is 9-3?., which expressed by words should be Kotim and titonhu. The tim hand goes round only once in a night and day, being on at midnight, and on S at noon. If a third index hand is added on the same centre, to represent the second hand on our ordinary watch, it should make one turn on the dial for each timsan, when the small division on the circle would indicate ilmlongs or y o|^-o ^^ part of the thn^ which is y-||o parts of our present second. Such delicate measures of time are often required in Physical Science, as in Astronomical observations, velocity of light and electricity, gunnery, &c. A further extension of delicate measures of time will be conceived in musical vibration, which I shall arrange into immediate connection with the tonal watch, that the base note for the natural key, shall make 10 (16 dec.) vibrations per timmill. Turn yourself towards the south with the tonal watch in vour hand, and it will be found that the timhand follows the sun nearly; or lay the Watch horizontally, so that the timhand points tow^ards the sun, and the figures on the dial will give north, 8 south, 4 east and F west, nearly. 34 1 io" 4 This is easily comprehended by the public, as the tonal compass, fig. 5, is divided the same way. A course noted from the tonal compass is clear and simple. One miUmeter in length on the equator corresponds with one timmill in time. By this division of time, it is always clear whether it is in the morning or even- ing, without any special notation. Our present system often leads to error or confusion, whether a noted time is meant in the morning or evening. 35 Fig. 5. Division of the Eartli's great Circle. The latitude or meridians should be divided from north to south into 8 tms^ with at the north pole, 4 tims at the equator, and 8 at the south pole. The equator to be divided same as the clock or compass. Nations ought to agree, to count the longitude from one meridian drawn through a fixed point on the globe. The different notation of longitude on maps is a great inconvenience and sometimes causes confusion. In my present traveling I have maps on which the 36 longitude is noted, some from Greenwich, some from Paris, Pulkova, Washington, Ferro, and on some maps it is not stated from where the longitude is counted. Independently of the different points from where the meridians are noted on maps, the present divisions of the circle make it very complicated to calculate the difference of time between places, and very few will understand how, — in fact the complication is such as to discourage many persons from the attempt ; while, if the circle and time were divided as herein proposed, the very figures denoting the meridians would give the difference of time by simple subtraction. In the Canary Islands appears to be a proper point to place the zero-meridian, as the ancient geographers who liave taken their first meridian from the west side of the Island of Ferro 17° 52' west from Greenwich. Maps constructed on such principle, would to our descendants forever indicate, not only the true posi- tion of the place on our globe, but the scale of latitude would give all distances on the maps in miles, (timmills) feet (meters) and inches (metertons) also the area in acres ; and a difference of latitude placed along a parallel, would give the correct distance corresponding with time in longitude. Those plain matters are by our present system, not only compli- cated in calculation, but are seldom thought of, for the complication screens away the simple knowledge. Measure of Surface. Tonal System. Old System. One square meter rr 0"239 square feet. 1 Square tonmeter rz 61'15 " 1 Square sanmeter r= 15658'768 " 1000 Square meters = 0-36 Akres. 37 The square sanmeter to be the measure of hind, corresponding to the acre. Measure of Capacity. Toual Systrra. 1 Gallsan = 10 Gallmills = 1 Gallton = 10 Gallsans 1 Gall = 1 Cub. Meter 1 Tongall = 10 Galls on System. 0*79 cub. in. about a table spoon. 12 62 cub. in. about a tumbler. 201-78 cub. in. about a gallon. 11 Bushel. about 30 cub. feet. 478-2 cubic feet. 17-75 cubic yards. 1 Sangall =10 Tongalls — 1 MilLall=: 10 Sanjralls = 1 Millgall ■= 1 cub.tonmeter = The Gall or cuhic meter to be the unit for measures of capacity, in ordinary market practice. The Scmgatl to be the measure of excavation and embankments, also for grain, &c. The Millgall to be the measure of firewood, being one c}it)ic tonmcter. Measure of Weight. One cuhic meter of distilled water will weigh 7'3017'4: pounds avoirdupois, to be the tonal unit for weights, and to be called a Pon. Toniil Sj'stem. 1 Ponmill 1 Ponsan 1 Ponton 1 Pon 1 Tonpon 1 Sanpon 1 Mill pon 10 Ponmills 10 Ponsans 10 Pontons 10 Pons 10 Tonpons 10 Sanpons Old System. = 0*0284:8 drams avoi. r= 0--45568 = 45568 pounds " = 7-3017 = 116 8 -= 1868-8 lbs. = 0-838 tons. = 13-34 tons. 38 The pressure of the atmosphere will be about 46 jwns per square metef\ and the height of a column of mercury balancing the atmosphere, about 5 meters. The force of gravity will cause a body to fall 3-:5-2?'7 meters in the first timmill in a vacuum, and the end velocity will be 72 562 meters per timmill. The Ponsan to be the unit for apothecary and minute weights. Pon for the ordinary market prac- tice. Sanpon as shipping unit and heavy weights, corresponding with the ton. Measure of Power. One pon lifted one meter in one timmill,, to be called one effect. By the present system, one pound lifted one foot in one second is called one effect, of which there are 550 effects on one horse-power or 55 effects on one man's-power. Tonal System. Old System. One effect z= 2-704 effects. i man's-power zz. 10 effects = 43'268 eff,=i 0-86 man. 1 horse-power =: 10 men zz 692*3 eff.i^ T25 horses. The mai'Cs'poiver to be the unit for manual labour, and horse-power for machinery and hea^'y work. Money. The American dollar is nearly the mean difference of all the monetary units of the world, and curious enough, compared with the largest the English pound sterling £, and the smallest, the French Franc F, the Dollar D, will be the mean proportion of the two. 39 L : D = 1) : F. or D := YTV. If the world could agree to adopt one unit for money, it seems that the dollars has a claim to be chosen as a standard. Tonal System. Old System. One dollar =10 shillings = One dollar =100 cent. 1 shilling = 10 cents = 6:5 cents. 1 cent = 0-39 cent = 2 centims. The inconvenience of the monetary decimal system is daily felt in the actual market practic, for although the dollar is divided into 100 parts, for which suitable coins (most of odd numbers of dollars and cents) are in circulation, the retail prices of most articles are fixed to suit the dollar divided into 16 parts. A drink of almost any description costs 6 or 6J cents, which is Jg part of a dollar. A ride in an omnibus costs generally 6 cents. Suppose an article bought for 6 cents, and paid with a quarter of a dollar, there will be 19 cents change, summed up by the following coins lOct. + 5ct. + 3ct. + let. r= 19 cents. This can reasonably be called an odd system of calculation, because there is nothing but oddity about it. By the to7ial money, the same article paid by a quarter, which would be 4 shillings, there would be 3 shillings change, in which transaction the mind was carried only to 4, while the decimal money w^as fumbling about among the odd numbers up to 25. 40 Tonal Clin?. United States Coin f 1 cent. Copper, { 2 cents. I 4 cents. Silver, Gold, 8 cents. 1 shilling. 2 shilling. 4 shilling. 8 shilling. 1 dollar. 1 dollar. 2 dollars. 4 dollars. 8 dollars. 10 dollars. 20 dollars. Copper, ^ 1 cent Silver, Gold, 3 cents. 5 cents. 10 cents. 15 cents. 20 cents. 25 cents. 50 cents. 100 cents. 1 dollar. 3 dollars. 2i dollars. 5 dollars 10 dollars 20 dollars The tonal coins are all of even and of the easiest countable numbers, such as are required in the market, while the decimal coins are most of odd numbers, and of a complicated composition for calculation, even the half dollar or 50 cent has a prime number to its index. The ional coins give a nicety of a^-Jg th part of a dollar, while the decimal coins give it only to j^-^ part. The legal interest on money, in most countries is about 6 per cent, which by the tonal system would be nearly 10 per sant ; consequently, calculating that interest on money, would be only to point off two figures on the capital. If the tonal interest is 1 more or less than 10 per sant, it is calculated by simple addition or subtraction. 41 Interest on 3^5c^4 dollars at 10 per sant. = 3^5-':54 (C (( (( (C a 11 ii = 3T0-^g4 which is 3T0 dollars, & shillings, and 8- cents. This makes a simple interest calculation in the neighbourhood where it is most wanted. The differ- ence of 1 "pev cent interest in the neighborhood of 6, is a rather large margin, for which we often find it accompanied with a fraction, in practice. One per sant tonal z=z O'o91 per cent decimal. One decimal per cent, is 2*56 per sant tonal. Fractions would be rarely required to the percentage in the tonal system. The most common retail prices of articles in Amer- ica are as follow : Market Prices ct. Tonal Shillings or 16ths of a Dollar. Nearest Decimal Cents. 6J 1 6 12i 2 12 or 13 m 3 19 25 4 25 31J 5 31 371 6 37 or 38 43f 7 44 50 8 50 ' 56J 6 56 62i 9 62 or 63 m « 69 75 19 15 81i 8 81 87J S 87 or 88 93| f 94 100 10 100 42 It may be remarked that those prices are retained from the circulation of Spanish Coins in the United States, to which I beg to reply that if such prices and coins were not the most natural to the mind, and the most suitable for the market they would not be retained. Postage Stamps. The following are the Post stamps of the United States. let., Set., lOct., 12ct., 24ct., 30ct., 90ct. The very first glance at this series shows plainly that there is some confusion about it. The stamps of even post prices are not even in a dollar (except let.) and four of them are not even in any coin, there is a 10 cent post stamp and no 10 cent postage. The simple and even numbers, most valuable in calculation, as 2, 4, 8 and 16, are of necessity omitted, because the decimal system does not admit the natural numbers. Let us now turn our attention to Tonal Post Stamps. Tonal Stamps. Ameru an Cents. 4 cents for city post — ly\ cents. 8 cents " single letters - 31 " 1 shilling " double letters - 6i " 2 shillings " quadruple letters - 12i " 4 shillings " 8 — 25 8 shillings '^10 — 50 1 dollar " 20 — 1 dollar. Here it will be found that the tonal pod stamp series 43 contains the even number most simple for calculation, and they are even both in post prices and in the tonal coins or dollar. Division of the Year. The new year ought to commence at Christmas, and the year divided into 10 (16 deci.) months, which would make about 17 (23 deci.) days per month. S E a o-a 3 1 2 3 4 5 6 7 8 ;S 9 % 10 8 10 '" a 17 16* 17 16 17 17 17 17 17 17 17 17 16 17 17 17 Names of the new months. Anuary. Debrian. Timander. Gostus. Suvenary. Bylian. Ratamber. Mesudius. Nictoary. Kolumbian. Husamber. Vyctorious. Lamboarv. Polian. Fylander. Tonborius. The first day of the new month to commeocu on. 21 December. 13 January. 4 February. 27 February. 21 March. 13 April. 6 May. 29 May. 21 June. 14 July. 6 August. 29 August. 21 September. 13 October. 5 November. 28 November. New year and Christinas. \ Night and day of ^ equal length. I Midsummer day, or St. John. \ Night and day of ^ equal length. * 17 Days in Leap Year. There will be 168 tonal days in a year, and 16S in leap years. 44 The names of the tonal months are given so, that the first syllable expresses the number of the month in the year, and every four months have a similarity in sound, impressing the quarters of the year. The new year and Christmas should be on the same day. There is no occasion for altering the days in the week, but when days are to be expressed by tonal fractions of the months or year, the number of days are nearly 80 per sant more than the fraction, for instance 6 days =r 0-4 months or 0-04 years, 6 days = 0*6 months, 19 days r= 0-8 months, T days =: 0-09 years, 12 days = O-OIO years, &c., &c. The different Kalenders used in different Countries, would by the tonal system at once fall into one. The old or Julian style is yet used in Russia and other Countries, it is 12 days behind our new or Gregorian style. The Evangelistic year com- mences December 27, on the day of St. John ; this style is also adopted in Freemasonry, where it is known as the Masonic year. The tonal sfyle would become seven days ahead of the Gregorian. Measure of Heat. The three different thermometrical scales causes a great deal of inconvenience in science and art. Al- though Fahrenheit's scale is generally employed in the United States, yet we have American Scientific books in which Celcius' scale is used exclusively. Celcius' decimal scale is the most convenient for calculation, but I believe that those degrees are too large for scientific purposes, that we want the scale to be divided into more parts between the freezing and boiling points. 45 By the tonal system it would become tlie most natural to divide the thermometer scale into 100 (256 decimal) parts between the freezing and boiling points of fresh water. Tonal System. Old System. Zero or = + 82 Fah. or Celcius. 1 Temp =r 10 tempton =z 11 J Fah. or 6 J 1 Tempton = 0-7 Fah. or 0-4 " Tenq) for rough measurement of the temperature of the weather, and tempton for scientific purposes. 3Iu8ic. The many different clefs used in music, seems to be a complication without remuneration. Music Corps often use four different clefs, namely, Bass^ Tenoi\ Treble and Alto^ all of which could be of one single denomination. The Tenor and Alto clefs are gradually withdrawn, but there is yet no indication of dispensing with the Bass or F. clef In Piano music particularly, it is an unnecessary complication, and burdens the student, to have a different denomination on each stave. I shall here arrange it so that all the different clefs will be represented in one denomination. The standard pitch of tone, I will assume to 100 (256 Arabic) vibrations per timmill, for the base note in the natural key. As A is the first letter in the alphabet, it appears natural that it should be the first or base note in the natural key, and that such an octave from A to A, should be located within the musical stave. 46 I would propose to denote five different clefs, as follows : CANTO CLEF. Soprano. — This clef to run two octaves above the treble pitch. ALTO CELF. Contralto. — One octave above the treble, for high female voice. TREBLE CLEF. XI Descant. — Natural position, or ordinary Y- female voice. TENOR CLEF. For the common voice of man, one oc- tave below the treble. BASS CLEF. - The ordinary bass, two octaves below the treble pitch. In running music, where the notes extend too high above, or too deep below, the stave, the octave can be altered, by placing the suitable clef on the bar where the change is required, similar to that now employed as 8!i. The following natural scales, in the different clefs, show the number of standard vibrations per timmill, of each note, divided according to the geometrical progression or tempered scale, as employed in practical music : 47 800 8 London, 31st October, 1859. \ Sir: — Your esteemed letter of the 1st of June, has been received, together with a copy of your description of the Calculator, which is evidently a most ingenious and useful instrument, 'and the manuscript account of your new system of arithmetic and measures, weights and coins. We think ourselves much honored by the confidence which you have manifested towards us, and are of opinion that we shall best testify our high sense of your ability and intelligence, and your zeal for the improvement of mankind by the most free and sincere expressions of our sentiments upon your project. The project has evidently the great merit, which, as far as we know, belongs to no other hitherto invented, except the metrical system, that is a uniform system founded upon one simple principle, which is con- sistently applied throughout to the attainment of its professed design. We are, nevertheless, sorry that we cannot give it our support, having by the very consti- tution of our society, and from its first foundation, adopted the number 10 as the basis for such a system. On reviewing the grounds of our original determination we see no reason to depart from it. When the metrical system was invented by the careful deliberation of the first mathematicians of the age, they studied the question by the arithmetical scale, and especially to take 12 as a basis, because that num- ber seemed to have in some respects a claim to be 63 taken in preference to 10. After a full examination of the question they decided that it was necessary to retain 10 as the basis in arithmetic, and to adopt it universally for measures, weights and coins. Your system would be far more difficult to learn than the other. When learnt, it would require a smaller number of figures in each operation, and might therefore present some facilities for making calcula- tions in writing, but it would be very burdensome to the memory so as to be unsuitable for mental arith- metic, and consequently for all the smaller dealings of the shop and the market, and for those minute calcu- lations which in all arts, trades and manufactures often requires to be performed with the greatest possible rapidity. There is a limit to the powers of the human mind, and it appears probable that, except in extraor- dinary cases a system founded on 16 as a basis, would be found to exceed the natural capacity of man for the use of numbers. You object to the meter, as " much too long to be convenient to the artizan, and you therefore choose for your unit a length which is about the seventh part of a meter. The proper aim in determining upon a unit of length is to find one adopted, as far as possible, to all uses without exceptions, and the general consent of mankind seems to point to the conclusion that a length approaching to the meter best corresponds with this intention. For specific purposes the meter is divided or multiplied either by 2 or by 5, and thus you may obtain any measure you please including your own unit which is very nearly equal to 15 centimeters. We could show you, if we had the pleasure to see you here, numerous decimal divisions of the meter such 64 as the measures 5, 10, 20, 25, 30, 40, and 50 centi- meters. We have these graduated down to half mili- meters, made of a great variety of substances, and with considerable difference of form, either solid and in one piece, or made to fold with hinges, or to be wound on roulettes in cases so as to be carried in the pocket with the greatest ease imaginable. Also some are made with slides. In short the meter is proved by experience, an experience which is extend- ing every day over wider and wider area of the earth's surface, to be adopted for the artizan as well as for every other occupation. You express a preference for the English foot, but if the foot has advantages, you may take the foot of Hesse Darmstadt which is exactly the fourth part of the meter. It is considerably nearer than the English foot to your own proposed unit, and in itself is unquestionably as convenient for the artizan as any other foot. In conclusion, we beg to present you with the prin- cipal publications which have been issued by our branch of the International Association. We would especially direct your attention to our treatise on the best unit of length. In section VI. you will find a discussion of the question respecting its adaptation to binary division, and in section VIII. it is maintained in opposition to your views, that the meter may be employed with the greatest possible advantage in the mechanical arts. You will allow us sir, to indulge the hope that further examination of the subject may induce you to coincide in the opinion which we endeavor to defend and which is gaining ground, as we understand, in Russia, as well as in the civilized countries. 65 Witli much respect we subscribe ourselves on behalf of the British Branch of the International Association. Your obedient servants, (Signed,) James Yates, F. R. S., Vice President. Leone Levi, Resident Secretary. Mr. John W. Nystrom, Rostof on the Don. P. S. — Further observations on the same topic are published in Lord Overstone's questions with the answers, London Folio, 1857, 176, 179. Your letter and pamphlet were shown at a meeting of the Inter- national Association, at Bradford on the 10th, 11th, and 12th ultimo, and we were desired to convey to you their thanks for the valuable suggestion you have offered. Leonf Levi, Res. Sec. ludinova, in the government of Kaluga, Russia, Nov. 26, 1859. To the International Association for obtaining a uniform Decimal System of Measures, Weights, and Coins, No. 10 Farrar's Buildings, Temple, London : Gentlemen : I have had the honor to receive your favor of the 31st of Oct'r last, for which I beg to return my most sincere thanks. Feel very much gratified indeed that your honorable body considered my suggestions worthy of notice. I hope the International Association will bear with me for making some remarks on your letter, by which 66 I have no other object, but to discuss the connection of practical and scientific principles, and sincerely beg you to pardon my straightforward expressions. It seems to me that the real substance of my project for the tonal system of arithmetic, weight, measure, and coins, is not well conceived or appreciated by the International Association, because there are statements in the letters which are not exactly in accordance with the fact, likely originated from the difficulty in con- ceiving a new arithmetical system with a new base, when the base 10 is impressed on the mind. You state that my " System would be far'more difficult to learn " than the other." Such is not the case ; it is easier learned, and although the multiplication table of the single figures is about two and a half the extent of that of the old system, it is much easier acquired. The only difficulty is, to turn the mind from the old base : for instance, 8X8 = 40, will appear very curious to a stranger, who is perfectly sure that 8 X ^ == 64, but when he knows that 16 is the base for the system, and 8 is half of the base, he will easily conceive that half of 8 is 4, and 8 X 8 = 40. " When learned, it would require a smaller number " of figures in each operation, and might therefore pre- " sent some facilities for making calculations in writing." This is of little or no importance in the sense you apply it : it makes very little diff'erence if one or two figures, more or less, are written down on paper, which is a mere mechanical operation compared with having the figures clearly located on the mind. Any measure under 8 feet can, by the tonal system, be expressed to a nicety of a fraction of a millimetre. 67 or less than 32nds of an inch, with only three figures, which, in ordinary cases, would require at least five figures on the French metre; and a measure of up to 122 feet is expressed to the same nicety by only four Tonal figures. The utility of the tonal system is not limited by the small number of figures expressing a delicate measure, but on account of the figures at the same time impressing the mind of the natural frac- tions, quarters, eighths, and sixteenths, the principal utility lays in the clearness of the expression. Six- teenths expressed by decimals will require four times the number of figures, which will carry the mind 1000 times further than the tonal system. In astronomical and nautical calculations, there will be required a less number of figures, on account of dispensing with a great number of tables. " But it would be very troublesome to the memory, " so as to be unsuitable for mental arithmetic, and "consequently for all the smaller dealings of the shop " and market, and for those minute calculations which " in all arts, trades, and manufactures often require to " be performed with the greatest possible rapidity." This is not right, and it proves that the utility of the tonal system is not within your comprehension. For mental calculations, the shop and the market, it is best suited, and the very reason why I have proposed it. For mental calculation in addition and substrac- tion, the mind need not be carried further than the base 10, and in multiplication and division only to 100. I would not take the trouble to invent or pro- pose a new system of arithmetic for " the greatest " mathematician of the age," to whom it makes little or no difference, if the base is a prime number, for his 68 ps, qs, and fs are applicable to any system whatever ; neither would I think it worth the while to alter the system of arithmetic for the small portion of the pub- lic who have to do with quantities only by pen and ink, for whom it is very easy to find a suitable system, and the decimal system with 10 or 100 to the base will answer that purpose fully ; but it is not so in the shop and in the market, where the natural fractions and aliquot numbers are wanted, as well for mental calcu- lations as for the mechanical divisions and proportions of materials. " There is a limit to the power of the human mind, " and it appears probable that, except in extraordinary " cases, a system founded on 16 as a basis would be " found to exceed the natural capacity of men for the " use of numbers." The decimal system is the worst that ever could be selected of the even numbers in the neighborhood of 10, eight or twelve would require less capacity of mind. It is easily found in practice that when the base of a measurement is 12 or 16, it is easier managed in the mind, even with the present system, and it would become so much easier if the Arithmetic had the same base. Had I proposed 11, 13, 11, 15, or 17, to the base, the system would become more difficult in a quadruple or triple proportion to the number of new digits added, but 16 is quite an exception to that supposition; 9 as a base would be a great deal more difficult than 10. I will here give you some numbers placed in order as they become more difficult as a base for Arithmetic, namely, 8, 16, 12, 10, 14, 9, 11, 15, 13. It can very readily be conceived that the present arithmetical base is too small, because, referring to the decimal system 69 in practice, it will be found that it is generally im- pressed on the people's mind that the unit is divided into 100 parts, by which quantities are generally writ- ten and expressed, for, although it is at the same time divided into 10 parts, it is seldom used so. For in- stance, in France, it is never said or written that a measure is so many decimetres long, but it is expressed in so many centimetres. In America, it is never said or written that an article cost so many dimes, but it is expressed in so many cents. We have also in America, from the Spanish money, the dollar, divided into 8 and 16 parts, which are mostly expressed by separate names, and in reality found to be a more suitable division for the market. Suppose an article to cost 38 cents, and it is paid with a dollar. Now, the seller must carry his mind to 100, and then back to somewhere about 70 ; here he will be confused about the eight, and not sure if it will be cor- rect to subtract the 8 from 70 ; but finally he finds out that it is 62 cents to be returned on the dollar; the buyer — most frequently not so smart in counting as the seller — will perhaps say that there should be 72 cents change. This example I have given from actual and frequent observation in practice. Now, suppose a similar example with English money : an article cost 38 pence ; it will be observed that 38 pence is not noted, but it is said or written three shillings and two pence. Suppose the buyer to pay for the article with a crown, which is five shillings. The seller will very likely reply, " Have you two pence, and I will give you two shillings'?" or he may give the buyer Is. lOd, and so the aff"air will end with perfect understanding ; the mind was not carried above the base 12, while in the 70 American case the mind was carried more than eight times further, namely, to 100. The decimal system is therefore very troublesome for mental calculation, and frequently approaches the " limit to the power of the human mind,''' which would be rarely the case with the tonal system. It will not be denied that halfs, quar- ters, eighths, and sixteenths are the most natural frac- tions for the artizan, shop, and market, and they are frequently expressed by decimal fractions; but if 0"125 is shown to the majority of the people, there will be comparatively none who understand the true meaning of it; and if it is told to them that 0*125 means J, it will be necessary to explain that the whole is divided into 1000 parts, and 125 of the parts is J of the whole. The people will then surely reply that this is a round- about way of doing things, and that they are not willing to cut their things up into 1000 parts in order to get it into eighths. I am inclined to believe that among the best arithmeticians, including the International Association, there are few, if any, who clearly compre- hend that 125 is J of 1000, but it is well known to be so by practice in calculation. It is easy to comprehend that 25 is J of 100, from which it can be conceived that 1 X 25 := 12*5, and by that way it may be im- pressed on the mind that 125 is J of 1000. If 12 was the arithmetical base, it is easily conceived that ^ = 0'16, but with the tonal sjstem it is most easy to comprehend that J = 2. Therefore the decimal system is very complicated and difficult, as well for mental calculation as for the artizan's ordinary appli- cation of numbers and measurement. A high number of several digits must be managed in the mind, in order to comprehend a small one of only one digit. 71 In music, the tonal system is in full operation ; the notes are divided, as regards time, into halves, quarters, eighths, &c., &c. Fig. 8. A bar of music is generally expressed by quarters or eighths, and a burden has generally 8 or 16 bars. Now, suppose that a musician is requested to divide his notes, bars and burdens into fifths or tenths, according to the decimal system, Fig. 6. thus r r p -^k\ /r^^ orifyouplease rf Jf X rrrrrrrm rrrrrrrrrr then ask the musician to play a decimal piece of music, and it will sound very much like the decimal system introduced in the shops and markets. This is the best comparison I can give between the tonal and decimal systems, because, if the world was fortunate enough to be in possession of the tonal system, and knew nothing about it, but requested to turn the mind towards the decimal system, it would be much more awkward to mankind than for the present musician to do so. " You object to the metre as much too long to be convenient to the artizan," — Yes ; and the Interna- tional Association has given me a further proof of the 72 fact, which I beg to explain hereafter, — " and you " therefore choose for your unit a length which is about " the eighth part of a metre." It is stated in the manu- script that the length of a step of a man, or about two feet, appears to be a suitable unit, and when I divided the circumference of the earth with 16^ it was my greatest wish to arrive at a unit of about 20 to 28 inches, but as the length of the assumed main standard was not under my control, I was obliged to be con- tented with the last quotient 5*865 inches. It was, however, my intention to propose to divide the quad- rant of the earth with 16", which will give a unit of about 23| inches, but in order to follow a uniform and unbroken system of division throughout all kinds of measurement, I concluded to maintain the first quo- tient of 5-865 inches as the unit for length. " The proper aim in determining upon a unit of "length is to find one adopted, as far as possible, to all "uses without exception." That is just the very object of my aim, and it is the inconvenience and defalcation of the decimal and metrical system, that has called on me to propose something better. It seems to me that in making such statements, it would have been well to give some reason and example where the tonal system is inapplicable. The metrical system is inapplicable in navigation, because it does not agree with the degrees and minutes of the great circle of the earth, which, also makes some inconvenience in geographical survey. The decimal system cannot well be adapted for the division of the circle and the time, nor can it be adapted in music, which forms the most natural conception of division. " The general consent of mankind seems to point to 73 " the conclusion that the length approaching to the " metre best corresponds with this intention." This is not correct. If the table of foreign measures of length is examined, it will be found that the whole world points towards the English foot, that the French metre stands alone the longest measure, and that it is only the Persian arshine which attempts to approach it. Nations point towards the French uniform decimal system, merely because it is, as far as our present arithmetical system permits, in itself the most complete for calculation, but if the French had adopted a shorter metre, I believe the system would have been picked up much sooner by other nations. " For special purposes the metre is divided or multi- " plied either by two or by five, and thus you may " obtain any measure you please, including your own " unit, which is very nearly equal to 15 centimetres.'' That I do not understand. " We could show you, if we had the pleasure to see " you here, numerous decimal divisions of the metre, "such as the measures of 5, 10, 20, 25, 30, 40, and 50 " centimetres." This is a proof that the metre is too long, and very likely some practical mechanic or engi- neer has made the same remark, for which the metre is cut up into pieces, in order to show that it can be made shorter. Let us examine the pieces one by one. A measure of only 5 centimetres is of little importance to the artizan, besides 5 is a prime number, which makes the whole decimal system objectionable. A length of 10 centimetres is very convenient for minute measurement, but too small for general use. Twenty centimetres is a good measure within itself, may be conveniently used in the drawing-room and for 74 measures not exceeding its length, but for more than 20 centimetres, it will be accompanied with an objec- tionable mental calculation. It is contained in the unit five times, which is a perplex number for the artizan, because when the fifth part is laid down he may be uncertain whether he has laid down four or five 20 centimetres, and when he looks back on the measured part, he cannot well conceive the correctness without going over it once more, Avhich would not be the case if the measure was contained four times in the unit, where the halfs and halves certify the correctness. A measure of 25 centimetres has the only advantage of being contained 4 times in the metre, but within itself an unsuitable measure ; in measuring off a distance between 25 and 100 centi- metres, it is accompanied with a troublesome mental calculation. Twenty-five centimetres are rather long for the pocket ; it must be folded, but into how many parts 1 if folded into two parts, there will be 12J in each. Thirty and forty centimetres are not evenly contained in the unit, and will, in practice, be accompanied with troublesome mental calculation. Fifty centimetres chopped up into four or five parts, has its evident disadvantages. If these diff'erent measures are introduced into the market, people will become accustomed, one to a 20 centimetre, another to a 25 centimetre, and some select 30, 40, or 50 centimetres; then when one gets hold of a strange centimetre, he is apt to make a mistake in his meas- urement and calculation. " We have these graduated down to half millimetres, '' made of a great variety of substances, and with 75 " considerable difference of form, either solid and in " piece, or made to fold with hinges or to be wound on " rollers in cases, so as to be carried in the pocket with " the greatest possible ease imaginable." I am perfectly convinced that the metre can be made convenient for the pocket, but I say that it is not convenient for measurement and mental calcula- tion, and I am sure that it requires a great many ingenious contrivances to put the metre into a suitable shape, but among all your varieties of metres and centimetres, have you a single sample which can practically be considered so good a measure as the English two-foot rule 1 You will allow me to doubt it. I have also in my possession a few varieties of the metre, but none which I consider a proper measure- ment, and I have never seen a good metre even in France, although I have made great efforts to procure the best possible. Those in my possession are all made to fold into 10 parts, made of ivory, bras^■, fish-bone, wood, and one a tape to roll in a case, but they are all toys. In Marseilles once I bought a metre of the ordinary form, made of ivory, to fold into 10 parts, went home to my hotel and tested the metre on my standard rule, and found it to be 1| millimetres too short. I returned immediatelv to the instrument maker, Mr. Santi, No. 6 Ferreol Street, stated the fact, which was soon testified on a standard metre, and I was offered to select a correct one, which made me try a great many metres one by one, and did not find two of the same length. I then suspected the great many joints, tried several by pushing and drawing, when I found a little motion in some of them, tried again two metres, the shorter one I stretched a little, when it 76 became the same length as the other. I selected one metre by the standard in the store, which I have now on my table ; it has grown two millimetres longer when I stretch it out, and when I push all the joints in an opposite direction the metre will be J millimetre too short, I do not blame the workmanship of the metre in question, because it is made as good as it can be, and it is equally good as those I selected in Paris, where I found similar metres to those in Marseilles; but I object to the principle of the instrument, because it is in every shape inconvenient in practice. It is very inconvenient to lay out work by the ordinary pocket metre ; for instance, the metre must be kept and adjusted by the left hand at a, and stretched by the Fig. 9. a 2r right hand at ^, it is then required a third hand to straighten the decimetres between a and 5, because the work is oftentimes such that if the hand is taken from J, that end of the meter will fall down, and disturb the adjustment at a. In practice it often happens that it is inconvenient to get at one of the points between which a measure is wanted, a two foot rule is then stuck over to the furthest or otherwise inaccessible point ; and the meas- ure read at the nearest point ; in a great many such cases of daily occurrence, it would be impossible to employ direct a ten-folded French metre, for which the two hands are required, one at each point. Suppose the outside diameter of a cylinder is to be measured, it is generally taken in a pair of callipers, then by the 77 English mode the callipers are kept in the left, and the rule in the right hand, while the diameter is read ; now by the French measurement two hands must be employed to keep the metre while a second person must be employed to keep the callipers. You will now surely remark that " the metre can be made to " fold with hinges into four parts — similar to the " English rule, and used with the same advantage" — to which I beg to reply that the metre in such a form will be rather clumsy for the pocket, and for the artizan, and on account of its great length it will not have the firmness of an English rule. Two and a half decimetre or the odd number of 25 centimetre in each part is an indication that there is something wrong about it. Half a metre folded into two or four parts is a broken up half thing — I say broken up, because two parts will contain each an odd number of divisions 25, and four parts, will contain each 12 J centimetres. Another measure generally employed as a standard by architects, city surveyors, in machine shops, &c., &c., about 8 to 12 feet long, and very likely in the office of the London City Survey will be found stand- ards of 10 feet, which is a very convenient measure for a great many out-door works ; a measure of that kind will be about three to four metres, which are very inconvenient numbers — accompanied with extra calculations in laying out a long measure for which a tape or a chain is not correct enough. If such a measure is made five metres, it will be rather long and inconvenient, and accompanied with a mental calculation which by the prime number 5 gives an odd number at every other operation. 78 A measure of 10 metres cannot well be employed in the streets, except in the form of a tape line or a chain, but for such form 10 metres is too short. A tape line or a chain ought to be about 50 to 100 feet long. Further you state that, " in short, the metre is " proved by experience, an experience which is extend- " ing every day over wider and wider area of the " earth's surface, to be adopted for the artizan as well " as for every other occupation." This is saying much. Has experience ever had anything to do with the length of the metre from its very first origin 1 It was according to your statement '■'invented hj ilie first '•'mathematicians of the age^' after which it was in- truded on the French artizan by law, from which experience in using it. was necessarily attained. The mathematician had the measure of a quadrant given to him in figures, which he found was easiest to divide by 10s in order to arrive at a small number, but had the mathematician been set into practice to divide a quadrant or a straight line by a pair of compasses, he might have discovered that the most easy, and the most correct divisions are attained by dividing it into half and halves, which would have given a quotient of about 23 J inches as a metre. The length of the metre has nothing whatever to do with the utility of the French uniform decimal system of weight, measure and coin. Had a shorter metre been adopted, and such a cubic metre of distilled water called a killograra, the same advantage would have been attained. The prin- cipal difficulty in introducing the decimal system, and the general discord of weight and measure throughout the world, is caused by the unsuitable base in the arith- metical system. 79 It would indeed be a great service to mankind, if some leaders of the scientific world would deviate a little from their determination to maintain and pro- pagate a system so ill-suited for the purpose for which it has so long toiled. " You express a preference for the English foot, " but if the foot has advantages, you may take the foot " of Hesse Darmstadt which is exactly the fourth part " of a metre." The Darmstadt foot would do me pre- cisely the same service as any other of the nearly 35 different foots employed in different parts of the world, I do not care how many times it is contained in a metre, because that has nothing to do with the subject in question. The object aimed at is to invent, propose and introduce to the world a system of calculation, weight, measure and coins, which would without excep- tion fulfil all requirements of mankind, and when such is attained, it makqs no difference how many times the Hesse Darmstadt foot goes in a metre. I thank you most sincerely for your kindness in offer- ing to me the principal publications issued by your branch of the International Association and hope soon to receive them, and I shall read them with the greatest interest. I am well convinced that you have furnished plenty of good materials in favor of the decimal and metrical system, which are current among a great many of your readers. You can give many instances where the metre can be conveniently employed ; you can give examples that when so many tons, cwts. and pounds is multiplied by so many £ sterling shillings and pence, and divided by so many fathoms, feet and inches, will be a long and complicated calculation, compared with the measures at once expressed by decimals ; besides 80 the metrical and decimal system being adopted and in successful operation, by one of the first empires in the world, is indeed a great temptation. According to your statement, I expect to find in " Section VIII., it is maintained in opposition to my "views that the metre may be employed with the " greatest possible advantage in the mechanical arts." Such is easily maintained in writing, but go to prac- tice, and give an English mechanic a French metre of the ordinary ten-folded form, and ask him to measure a distance of about 20 inches ; the mechanic will then fumble about in straightening the decimetres, and if there is no support between the two points, he will hang the metre in a catenary form, as he is not accus- tomed to employ two hands for such a small measure, — he will then very likely tell you that the measure between the points is 50 and some small marks which he cannot read. Now give an English 2 foot rule to a French mechanic, to measure a similar distance, and he will tell you immediately without hesitation that it is 20 and f . The inches being divided into halves, quarters and eighths, makes the reading so clear, that the very first glance impresses the mind of the correct measure. A captain sailing along the sea cost in a dark night, requires to be if possible always in sight of a light, in order to be sure of his position, and safety of his ship ; such is the case with the mind sailing along a graduated measurement. On the English rule, fig. ^', there are big lights at short intervals, and beacons and buoys between them, while in the wilderness on the French metre, fig. 1^, you encounter sometimes a little light high up in the arithmetic atmosphere, looking very 81 Fig. U. 7 III' 8 I I I ±j_^ 9 much like the old street oil lamps before lighting gas was invented, and between them you encounter a num- ber of things one like the other, by which you are not Fig. V. m a 5 sure whether you are here or there. The ordinary English rule such as made by Mr. Elliott, London, or Field and Son, Birmingham, will stand and measure as long as twenty French pocket metres, and it will measure the last piece as correct as the first one, which is not the case with a ten-folded metre stretched a few times. On mathematical instruments in general, the deci- mal division is very troublesome, compared with the natural divisions, for instance, in verniers, fig. 8, this makes a clear reading, and divides the inch into 256 parts, while the decimal system, fig. ^, is more difficult and divides the inch into only 100 parts. Fig. 8. Fig. S. 1 1 1 1 1 1 1 1 1 j_ TT 7 \ I / 1 1 1 1 1 1 1 1 1 III 1 1 1 b: I I 1 I .1 I I I I I I I i~rr 1 j\ I regret very much to say that the closer I examine the subject, the more I am inclined to oppose the 82 French metre, as well as the decimal system, which is in reality the most unnatural system of division, which could reasonably be selected. I am sure that in a thorough practical examination the metre will stand a poor chance, and I shall be much mortified if the law intrudes upon me such in- convenient measurement for my mechanical works. Weight. The decimal system is equally inconvenient for weight as for all other measurements, the unit being divided into 10 parts, for which are required -five different weights in weighing all the ordinal parts namely 1, 2, 3, 5, and 10, or a weight of 4 may be substituted for the 3, but it is at any rate an odd and dreary composition of weights. 1 = 1 weight. 2 = 2 weights. 3 = 3 " 3+1=4 " 5 = 5 " 5+1=6 " 5+2=7 " 5+3=8 « and 5 + 3 + 1=9 " thus all the ordinal parts of 10 can be weighed. Now suppose a similar example with the tonal system, which will also require five weights, namely, 1, 2, 4, 8, and 10, this is the most natural composition of weights, they are convenient in the operation of weighing and easy for mental calculation. 83 1 1= 1 -weight. 2 n: 2 weights. 2+1=3 4 — 4 « 4 + 1 =: 5 4 + 2 = 6 4+2+1=7 8 = 8 8 + 1 = .1 8+2=9 " 8 + 2 + 1 = ^ « 1 + 4 = '(9 " 8 + 4+1 = 8' " 8+4+2=2 and 8 + 4 + 2 + 1 = T " thus all the ordinal parts of 10 (16) can be weighed. It will be observed that the five decimal weights could weigh only the 10th parts of the unit, while the five tonal weights give a nicety of every 16th part ; consequently the tonal system has in that case 60 per cent, advantage of the decimal system, and moreover the tonal weights give the natural and desired fractions, quarters, eighths and sixteenths, which is not the case with the decimal weights. For the natural fractions it will require three more parts to the decimal weights, namely |, ^ and ^, or expressed by decimals it will be 0*5, 0"25 and 0125, by which the sixteenth parts can be weighed, but it will be a complicated expression, for instance, 6 parts will be expressed by 0*375 and 7=0*4375, which can never be clearly comprehended, because the mind must be carried away to several thousands for only one fio-ure. 84 The decimal system can never avoid the expression of the tonal or natural fractions, because they are of daily occurrence in practice, while the tonal system is complete in itself for all uses without exception, and needs no reference to, but will do best without, the decimal system. If three more parts are added to the five tonal weights, namely, 0*2, 0-4 and 0'8, it can weigh to a nicety of every 128 parts of the unit, the expression will have one decimal (called a tonal) by which the true weight is clearly impressed on the mind. If you examine all the papers that have been written on the subject in question, including your own, and collect all the advantages and disadvantages of all dif- ferent systems in your memorandum, then examine well the tonal system, and you will find that all your collected advantages are contained in the tonal system, and all the difficulties and disadvantages are overcome. Your most humble and Most obedient servant, John W. Nystrom. On my visit in London I had the pleasure of meet- ing James Yates, Esq., M. A., F. E,. S., Vice President, and Professor Leone Levi, F. S. A. F. S. S., resident Secretary of the International Association. Mr. James Yates was so kind as to invite me to his house to see the great variety of French metres spoken of in the preceding letters, which was indeed a fine col- lection. The best form of the metre in the collection, and the one best suited to the artizan I believe is the four folded one. Among the ten folded metres was 85 found what I have before remarked, none of the same length, but they differed up to IJ millimeters. Other forms, parts and divisions of the metre did not however alter my views, but rather strengthened my opinion herein given on the subject. John W. Nystrom. London, September, 1860. , Esq., President of the Society, Philadelplda. Sir : — I have left in the care of Professor X. a manuscript on a new system of Arithmetic, Weights, Measures and Coins, intended to be submitted to your consideration for publication. All the engravings and types for the new figures are ready for the press. In it you will find some corres- pondence with the Decimal Association in London, which is believed worthy of publication for the argu- ment on the French metre. Yours, most respectfully, John W. Nystrom, 1216 Chestnut St., Ph'da. Philadelphia, Se2'>. 25, 1861. Philadelphia, Oct. 19, 1861. Mr. Nystrom : Dear Sir : — I regret to announce that the report of the Committee on your essay that it recommend that the essay be not published, was adopted by the Society at its meeting, last evening; and the MSS. was ordered 86 to be deposited in the archives of the Society, subject to your order. Professor A. and Professor B. afterwards discussed your tonal system, and Dr. C. the octonal system of Mr. Taylor of this city, whose pamphlet was laid on our table, and seems not to have been noticed by you.* It was suggested that it would be agreeable to publish some abstract account of your system in the running minutes of the proceedings of the meeting. Very respectfully, X., Secretary. Philadelphia, Oct. 11, 1861. Prof. X., Secreiary^ c&c, &c. Dear Sir: — I herewith return the MSS. of Mr. Nystrom, to be examined by the other members of the Committee. I believe no other report will be necessary than simply to recommend for publication, or the contrary. Although Mr. N.'s papers have failed to convince me of the great gain by substituting the sexta decimal (" tonal") basis of notation, for the decimal, yet it is interesting and instructive, to have such a system fully worked out, and placed before us in all its bearings ; that is, when it can be done by a philosophic and com- petent mind, as is manifested in the case before us. I would, therefore, be in favor of publication, at least as far as page 60, which concludes the main recital. The remainder, which is of equal bulk, is a correspondence between Mr. N. and the officers of the International Decimal Association, at London. To my own appre- hension, there is some defect of force and perspicuity * I was not there. — N. 87 in their criticisms, affording Mr. N. the opportunity of making pretty sharp replies. All this, while it tlirows light on the subject, and is spicy enough to aid in the digestion, may be considered as somewhat of a repeti- tion. I have pencilled down a few random comments, and have had them copied on another sheet; and if you please, would like them, with this note, to be handed to Profs. A. and B., along with Mr. N.'s book. I conclude by proposing that the Committee, meet at the hall on the evening of the next meeting of the Society, 18th inst., at a quarter before 8 o'clock, to determine their report. Very truly, yours, D. Hashj comments on Mr. Ntstrom's new hasis of Arithmetical Notation, (Page 8, et passim.) {First comment.) The term "binary division" suggests the neces- sity for coining a new word. Binary refers to a douh- ling, not a halving process. Demidial or dimidiary, (from dimidium half) would express the very idea ; but as yet there is no such word ; nor any that expresses the idea. Inasmuch as Mr. N. finds it necessary to make many new words, this one is respectfully offered. (Page 21.) {Second comment.) If this new system would afford a relief from endless fractions, it would be a triumph over the decimal system ; but it does not. While a sixth part is represented in the decimal system by -16666 . . . forever, it stands in the tonal system, -29999 . . . forever. It works well for halves, eighths, sixteenths; but does not work at all 88 for thirds, fifths, sixths, and so on. Yet these divisions are continually occurring in practice. (Page 40.) {Third comment.) Prices are of every imaginable figure. A car ride is five cents. An exchange ticket seven cents : a pound of sugar, 9, 10 or 11 cents. Mr. N.'s system would be so much bothered by these, that he would probably insist that prices should be such as to make the work- ing easy. He is quite in error about the dollar holding a medium place among the monetary units of the world, and therefore having " a claim to be chosen as a standard." In calling the French franc the smallest unit, he forgets the piastre of Turkey, the rial of Spain, the drachm of Greece, and some others. Nor is the £ sterling the largest unit; there is the milreis of Portugal, and of Brazil. These errors, however, are not material to the merits of the scheme. (Page 57 to 60.) {Fourth comment.) The account of the Russian stchoty or counting machine is interesting. It is essentially the abacus of ancient Rome, and of modern China and Japan, the apparatus of a people very low in the scale of mathe- matical science. Yet Mr. N. would have it brought in our schools and counting houses, to help the new tonal system, and " turn the mind from the old basis." It would surely be a retrogade to put away the slate and pencil for this machine. (Page 69.) {Fifth comment.') If Mr. N. had observed the practice of our market people, he would have found his argument against the decimal system materially weakened. An article costs 38 cents ; the buyer hands out a dollar ; the seller in making change, is sure to act thus : — first lays down 2 89 cents, to bring his mind to 40 ; and then easily makes up the remaining 60 with a half dollar and a dime. So that he first steers for the nearest ten to rest upon, and from that completes the operation. He never thinks of mentally subtracting 38 from 100, unless he be an old accountant, or schoolmaster. It may be observed that Mr. Alfred B. Taylor of this city, constructed an ingenious system on the octonal basis. Mr. Pitman, the celebrated phono- grapher, urged a duodecimal reform ; and Dr. Patterson used to mourn that our arithmetic was not based upon 12 instead of 10. PkUadelphia.) 1216 Chestnut street^ Oct. 23, 1861. Pkofessor X., Secretary &c., dec. Dear Sir: — Your favor of the 19th inst. is at hand, I am sorry to hear the Society did not deem my manuscript worthy of publication. It is true I have not noticed Mr. Taylor's octonal system, as my tonal system was written in Russia long before Mr. Taylor's octonal system was published in America, and even if I had seen it, it would not have altered one sentiment in my manuscript. The Society " suggested that it would be agreeable to publish some abstracts," which I suppose from Mr. D.'s letter to you dated Oct. 11th, would be to omit my correspondence with the International Decimal Association in London. "When my Tonal system is published, I shall omit nothing of the manu- script, even my correspondence with and remarks made by the Society will be published, as I desire to have the subject thoroughly ventilated. 7 90 It may be found that there ''are some defects of force and perspicuity in" the comments made by Mr. D. which affords me a second " opportunity of making " a pretty sharp reply." The first comment is "The term binary division sug- •' gests the necessity of coining a new word." I have only to refer to page 61, where the International De- cimal Association in London, uses the same expression. " Binary" can be applied to halving as well as doubling, the difference is only to go up or down the steps. The word " Binary Divisions" is freely used in Mr. Taylor's report on the octonal system. A binary com- pound, say chloride of sodium, Na, CI, contains half of each substance, which is an example of a binay halving process. Second comment. " If this new system would afford " a relief from endless fractions^ it would be a triumph " over the decimal system, but it does not." It would indeed be a triumph ! but Mr. D. will never be satisfied on that point, for let us even propose one system of arithmetic for each fraction, or attempt to invent a system of arithmetic that would have no prime num- bers, he will still be disappointed. Can Mr. D. describe a circle through these four points ..'% and it will be a triumph in geometry. " While a sixth part is repre- "sented in the decimal system by 0"16666, forever, it " stands in the tonal system 029999, forever. It works " well for halves, eighths, sixteenths ; but does not work " at all for thirds, fifths, sixths, and so on, yet these " divisions are continually occurring in practice." Fifths are generally employed for the necessity of accommo- dating the decimal system which imposes so much incon- venience upon us. Sixths and 12ths are often used in 91 practice as an improvement on, or to avoid Sths and lOths; with the exception for the circle, 6ths and 12ths are of little importance compared with the binary frac- tions. In the following table are set down the fractions in question, with one, two, and three, decimals with their errors, in the Tonal, Decimal, and Octonal systems. Systems. One Decimal. Two Decimals. Three Decimals. Fraction. Error. Fraction. Error. Fraction. Error. Tonal, Decimal, Octonal, Tonal, Decimal, Octonal, J =0-1 1=0-1 >=0-5 1 = 0-3 J = 0-2 0-0416 0-0666 0-0416 0-0208 0-0333 0-0830 i = 0-09 1 = 0-16 J-0-12 i-0-55 1-0-33 i-0-25 0-0026 0-0066 0-0104 0-0013 0-0033 0-0280 - =0-299 0-00016 6 1=0-166 0-00066 1 0-125 0-00065 ^=0-555 0-00008 i = 0-333 0-00033 1=0-2520-00130 It will be seen in this table that the fraction l ex- pressed by one decimal has by the tonal system 60 per cent advantage in the correctness over the decimal sys- tem. With two decimals 250, and with three decimals 410 per cent, advantage in the correctness. Still Mr. D. says these fractions "does not work at all." For thirds the favor is still greater for the tonal system. Third Comment. "Prices are of every imaginable " figure. A car ride is five cents, an extra ticket 7 " cents." In my manuscript I speak about omnibus prices, as they were when I left America for Europe in the spring of 1856, which was written in Russia in the year 1859, when I knew nothing about the street rail- road arrangement. Six cents or rather 6J is a very general price for articles, as being -j^th part of a dollar. After my return from Europe, I made the following observations on car ride prices : 92 Frcmhford and SoufhivarJc Passenger JR. li. Co. FARES. Southwark to Front and York streets, 5 Cts. Frankford, 10 Germantown road to Frankford, 7 Southwark to Episcopal Hospital, 7 Berks street to Harrowgate, 5 Frankford to Hart lane, 5 For Children under 12 years, 3 By examining this price table we find that all the prices are not only odd but of prime numbers 3, 5 and 7. The base price for a car ride is 5 cents, and for long distances double price, 10 cents. For interme- diate distances such as from Germantown to Frank- ford, and from Southwark to the Episcopal Hospital, is charged 7 cents, showing an attempt to charge a price half way between 5 and 10, which should be 7| cents, bat our coins as well as our decimal base does not permit such division, for which we must be con- tented with the prime number 7. It is a general custom over the world, to charge half price for children, which in this case should be 2| cents, but as we have no such coins, it is made up to 3 cents. Let us now see what the to?ial prices would be. By the tonal system it is very likely that the base price for a car ride would be 1 shilling, (6^ cents,) but sup- pose even this to be too high, and the exact value of 5 cents is required, which would be (9 tonal cents, the price table would be as follows : Southwark to York Street, Frankford, Germantown rd, to Frankf 'd, Berk street to Harrowgate, Children half price, Decimal. Tonal Prices. 5 Cts. '19 cts. 1 s. 10 1-8 s. 2 i 7 1-2 1-8 5 C? cts. 1 3 6 8 cts. 93 The tonal price in both cases are all of even and easy countable numbers, and divides the prices as de- sired in practice. The Sanford's Opera bill says : Admittance, - - - 25 cents. Half price for children, - 13 " Pennsylvania Railroad trains leave Philadelphia at : Mail train. Fast line. Through express, Harrisburg train, Dpcimal. Tonnl. 8 A. M. 11-30 A. M. 10-30 P. M. 2.30 P. M. 5-8 T. 6 9-9 In this time table there is no confusion of A. IM. and P. M. in the tonal column, but the correct time is ex- pressed by the fewest possible numbers, clear to the mind at the first glance. T means the hour mark. The original meaning of A. M. and P. M. is not generally known further than that it means forenoon and afternoon, but even that is sometimes confused. Two coal-miners, Jack and Harry, arrived at a rail- way station in England, and examined the time table, when the following conversation took place : Jack. I say Harry, what does P. M. mean % Harry. Penny a mile, to be sure. Jack. What does A. M. mean, then \ Harry. Oh, that must be a A penny a mile. Jach. Then we will go by the A. M. line. Jack and Harry were not acquainted with ante meri- diem and jpost meridiem. In the crowded railway guides it is often difficult to find out whether a noted time means in the forenoon or afternon, and it is often necessary in tables to leave a separate column for A. M. and P. M. 94 Rev. E. Barnham preaches in the Concert Hall, Philadelphia, every Sunday at: 10 J A. M., 3 P. M., and 7 J evening. Tonal time, 7 9 8 T. Our present system employs tivelve characters and one word, where the tonal system uses only three characters and the hour mark. The Pev. gentlemen seems disposed to divide his time as in the tonal system. Third Commemt. " Mr. Nystrom's system would be so " much bothered by those that he would probably insist " that prices should be such as to make the working easy." Every shop keeper attempts to arrange his prices into easy countable figures, but the inconvenience of the decimal system is so great that it is difficult or rather impossible to satisfactorily attain that object, as is readily seen in store windows by prices marked on articles 37| cents, 81^ cents, &c., &c., an attempt to approach the tonal system. If Mr. D. will take the trouble to examine the shop practice, he may discover that prices are already arranged to suit the tonal system, in spite of the bothered decimal coins. Fourth Comment, " He is quite in error about the " dollar holding a medium place among the monetary " units of the world." How does Mr. D. know that the dollar is not holding a medium place, among monetary units '? Did he try it % And if so, why not favor us with his result? Is it more or less] The following table contains the present monetary unit in most parts of the world. The table can however be varied by judgment of different units existing in different countries, that any one who feels disposed to make comments on it 95 has an extensive field to operate upon. When I first worked out this medium monetary unit the result came much nearer the dollar than in this table. Monetary units in most parts of tlie world. South American & > Mexican dollar, S Chili and New > Granada dollar, S China dollar, Austria, Bohemia > & Bavaria, Florin, S Denmark, Specidaler, France, Belgium, Switzerland and Italy, Franc, Great Britain, £ Sterling, India, Rupies, Hamburg, Mark, Hanover, llialer, Holland, Florin, Prussian, Thaler, Portugal, Millrea, Pome, Scudo, Pussian, Ruble, Saxony, Thaler, $. cts. 1-05 0-96 1-43 0-50 0-96 0-18 4-86 0-46 0-30 1-10 0-41 0-72 M2 1-04 0-77 0-63 Spanish, Piaster, Sweden, Riksdaler, Turkey, Piastre, Wirtemberg, Thaler, Total, Divided by 20 will cents as a medium unit. The main money of Europe is Pound $. cts. 106 0-27 004 1-00 18-86 oe 94 Sterling, Florin of Germany, Thaler of Germany, Puble of Pussia, Franc of France, Piksdaler of Sweden, 7-23 : 6 = 1-20 0-94 2 I 2-14 4-86 0-45 0-70 0-77 0-18 0-27 7-23 1-07 dels. Which can be considered a medium monetary unit of the world. Fifth Comment. " In calling the franc the smallest " unit he forgets the piaster of Turkey, the real of Spain, " the drachm of Greece, and some others. Nor is the " £ sterling the largest unit ; there is medries of 96 "Portugal, (§1 12) and of Brazil." I beg to assure Mr D. that tlie units referred to are not forgotten. If we go to those extremes it is difficult to know where to begin or end. We may call the American eagle 10 dollars a unit, the Napoleon 20 franc. Impe- rial 100 fr, doubloon of Spain, Central and South America, about 15 dollars. Russian imperial 5*15 Rubles. Dobras of Brazil 34 dollars, and other monetay units rapj^ins: from a fraction of a cent towards 50 dollars. If we go further to the corners of the land among the Esquimaux's or the Calmucks we may find a liide of any animals or a heap of hay to be a unit for trading. The £ sterling and franc are the extreme units known and handled over the whole world, in preference of which the outside units of Greece and of the Esquimaux could not be admitted in a general statement. The French Franc is used in Belgium, Switzerland, Italy and Algiers, by a popu- lation of about 70 million ; the £ sterling is used by a population of I suppose 50 millions. The franc and £ sterling put together, I believe would exceed all the rest of the money in the world. Mr. D. says " These errors however are not mate- rial to the merits of the scheme." I have not been able to find a single expression in Mr. D.'s comments that have any bearing whatever on merit or folly of the tonal system. Sixth Comment. " Yet Mr. Nystrom would have it " (the counting machine) introduced into schools and " counting houses to help the new tonal system and " turn the mind from the old basis. It would surely " be a retrograde to put away the slate and pencil for " this machine." I would like very much to see Mr. 97 D, with a slate and pencil alongside a Russian with a tshoty, I would give the example for calculation, and Mr. D. would soon find the utility of the Russian tshoty, and the retrograde of his slate and pencil. When the tonal system is well- acquired, the counting machine would be found superfluous, but for the first acquirement the slate and pencil would not answer the same purpose as the counting machine. Seventh Comweni. " If Mr. Nystrom had observed " the practice of our market people, he would have " found his argument against the decimal system " materially weakened." After having read Mr. D.'s hasty comments in the Archives of the Society, I proceeded up Chestnut street, when opposite the Masonic Hall I observed at the northwest corner of Eighth and Chestnut, at the doorway of Sharp- less' store, a pile of dry-goods upon which was a paper sign marked with figures as big as my hat, "Extra quality 87 J cts." (per yd.) Arriving at the southeast corner of Eighth and Chestnut streets, I saw at the Eighth street doorway of the same store, two piles of dry-goods marked one 37| cts. and the other 62| cts. (per yard). Looking up Eighth street I saw at the northeast corner of Eighth and Zane streets, in the doorway of a store, a crinoline in the inside of which was a paper sign marked 37| cts. I walked up to this store where 1 found in the window fifteen articles marked with the following prices, ^\ cts. 12J cts. 18f cts. 25 cts. 31J cts. 371 cts. &c., &c., all arranged to accommodate the easy counting as in the tonal system. Returned and went into Mitchell's restaurant, 808 Chestnut street, where I took a cup of coffee and 98 cakes; was handed an ivory ticket upon which was engraved the number 19, indicating the price of my refreshment. At the counter I inquired why they charged the odd number 19 cts. and was answered " It ought to-be 18| cents, but as we have no such " coins we make it even to 19." I then asked, would not 20 be a more even number] And was answered, " 20 is not even in a dollar." Here you will find that the prime number 19 is called even, and the even number 20 is considered to be odd. It is literally true, that 20 is odd in 100, and that odd numbers as 5, 25 and 75, are even in 100. Upon further inquiry of their prices, I was shown to the other side in the store, to a box of about 13 inches square, divided into 36 compartments, each con- taining ivory tickets marked with the following prices MitchelVs Price Ticket Box. 6ict. 13 19 25 31 38 44 50 56 63 69 75 81 88 94 100 106 1 112 ' 118 125 131 137 144 150 : : j 162 168 175 181 200 275 281 287 300 306 312 1 318 1 1 99 Here you will find that it is attempted to arrange the prices to suit the dollar divided into 16 parts, as in the tonal system. You will observe that most of the prices are of odd and prime numbers, and to make them perfectly correct, most of them ought to be ac- companied with fractions. It is evident that those prices are considered easy to the mind in the market, and how much better would it not be, if our arith- metic was based on the same principle ; every coin proposed in my tonal system agrees correctly with those prices. Taylor's saloon in New York, and a great many other establishments, have similar arrangements of prices. I left the restaurant, walked up Chestnut street, stopped at the store of Le Boutillier Brothers, No. 912, where I found prices marked in the window 62| cents, 87| cents, 75 cents, &c., &c. At Besson & Son's mourning store, 918 Chestnut street, I found in the window marked the following prices. De Laines, 12J cents. Reps Anglais, 37J cents. Cravellas, 25 cents. Mousselin, 6^ cents. De Laines, 18,| cents. Other articles, 62 J cents. Grandrill, 31 J cents. One article, 44 cents. No price had any indication of decimal division. Went home to my house 1216 Chestnut street, where I pay for my washing 62J cents per dozen. About two weeks ago, I was charged by a shoemaker 37 J cents for mending a pair of boots. AVill Mr. D. yet think that my observation of market practice would materially weaken my argument 100 against the decimal system'? and in case he suppose that this is my first attention to market practice, I beg to remind him that in my manuscript it is plainly stated that it is my observation of the inconvenience of the decimal system and arithmetic in the shop and market that has led me to propose the tonal system. In Mr. D.'s argument on the 38 cents, he still brings the mind to the high numbers of 40, 60 and 100, where the tonal system would bring it only to the base 10. The price 38 cents would be 6 shillings tonal. . The following table contains the most common market price. Market Prices in Cents. Tonal Shillings or IGths of a Dollar. Nearest Cents as Mitchell's Ticket-. 6i 1 6 12i 2 12 or 13 18.1 3 19 25 4 25 , 3ii 5 31 37J 6 37 or 38 43i 7 44 50 8 50 56J 9 56 62i 10 62 or 63 681 11 69 75 12 75 81i 13 81 m 14 87 or 88 93| 15 94 100 16 100 Can Mr. D. discover any utility in the centre column of this table, compared with the two outside ones'? 101 Lastly, Mr. D, says : " It may be observed that " Mr. Alfred Taylor of this city, lately constructed an " ingenious system on the octonal basis. Mr. Pitman, " the celebrated phonographer used a duodecimal reform ; " Dr. Patterson used to mourn that our arithmetic was "not based upon 12, instead of 10." It seems from these statements, that Mr. D. has no preference to any one of the three basis, 8, 12 and 16, that if I had proposed 14 as a base, he may have given it the same consideration ; and if such is the case with the Society, I do not wonder at all, that the tonal system was rejected for publication. There is nothing new in merely proposing a better base for our arithmetic ; that I believe has been done since the time of Charles XII., of Sweden, by hundreds, and been thought of by thousands ; for any self thinker with good reason of mind, sees plainly the foUy of our deci- mal arithmetic. I am surprised to find so many of the first leaders of the scientific world to be so short sighted, as not to see the inconveniences, but propa- gates a system so unnatural in all its bearings. About a year ago there was an Italian in London, who proposed the duodecimal system, but in no case have I found any of such systems worked out with examples into a practical shape. Most of the propositions have been made by mere scientific men, who have given excellent accounts of the history of arithmetic, and finished by merely proposing a better base. I believe myself to have commenced and continued from where they ended, and I suppose you to know what they have said. Purely scientific men are not the proper persons to handle this practical subject, for the decimal arithmetic 102 is so clear to. them, that they manage the figures and come to their results as easy as a musician who plays the crank organ. Their lack of direct application of their science to practice, screens away the real inconvenience of our decimal arithmetic, which is readily proved by feeble remarks frequently made by such men. Many of them confine themselves more to style of language than to the substance of the subject. It is not sufficient merely to propose or say that 8, 12 or 16, would be better as a base, but in order to make a clear and correct im- pression of its utility, it is necessary to enter into details with examples, that any one may be able to estimate its advantages without taxing his own mind. Still the nature of the subject is such, as to be apt to be called curious at the first glance. The octonal system has two serious objections: First. That the base 8 is too small. Our experi- ence with the decimal arithmetic is, that 10 is too small as a base. Secondly. As we progress in this world, generation after generation, we require larger and larger numbers in our transactions. That which Moses counted by thousands, are by us counted by millions, and I ven- ture to say, that with the present decimal arithmetic, there are very few who have a clear conception of the immense number of one million ; and the more com- plication we have to lead us to such a number, the more cloudy it will be to the mind. The immense numbers necessary in astronomy, expressed by decimal arithmetic are inconceivable, while the toned system gradually leads the mind towards infinitum. In the octonal system we have two figures already at 8, three at 64, and four at 512 ; while in the tonal 103 system we have two figures first at 16, three at 256, and four at 4096. One decimal million expressed by odonal arithmetic will be 3,641,100, and by tonal arithmetic 94,240, which is a difference of two figures. Also for decimal fractions the odonal requires more figures than the ional system for the same nicety. The duodecimal has many advantages over the decimal system, particularly in thirds and sixths, but this is overbalanced by the serious objection of it not admit- ting binary division to infinitum. Sixths and thirds work much better in the tonal than in the decimal system, as seen in the table, page 91. Many self-thinkers express their regrets that the arithmetical system was not from the beginning founded on a better base. Many of them, I believe, prefer the duodecimal system, and some express their wish that man would have had six fingers on each hand, which might have led to a duodecimal system. The Sixdiopt Family, in Central America, have six fingers on each hand, and six toes on each foot; they might have had accomplished that object. By this theory, I would, of course, prefer eight fingers on each hand. I know no tribe of people that can accommodate me, but am satisfied that five fingers will answer for the tonal system. Should we now succeed to introduce a duodecimal system, our descendants tvoidd surely wish for the toned system of 2cs, as we wish for a duodecimal system from our ancestors. If the Arabic notation employed in our present decimal arithmetic had been suggested to Moses when he wrote the ten commandments on Mount Sinai, he w^ould surely have made similar remarks as that made 104 on the tonal system, that such curious looking diardciers could not be understood by his people. My manuscript on the ional system has been sent to a great many places for publication. The Franklin Institute thought it would have a very serious effect on the number of subscribers of their journal ! The Smithsonian Institute would not publish it, because they had so much of the same kind before ! ! ! The U. S. Coast Survey stated they would publish it if recommended by a member of Congress.* And lately the Society of Philadelphia has rejected it, per- haps on the remarks herein replied to, but Mr. D. recommended its publication. I return herewith the pamphlet on Mr. Taylor's octonal system, and thank you very much for calling my attention to it. It is indeed an interesting and ably written work. I suppose I must follow the track of Mr. Taylor, and go to Boston to get my tonal system understood and appreciated. The heading of this letter is dated October 23d, when I intended to write but a few lines, but when I got into it, I could not well cut it off until it reached nearly twenty-four pages. It is now the 28th of October. Your humble and obedient servant, John AV. Nystrom. * The idea of showing to a member of Congress this manuscript and calculations, with Vs, ^s, &c., among the figures ! he would surely pro- nounce me a funny fellow. When scientific men, as at the Franklin and Smithsonian Institutes, Society, and others, cannot appreciate the subject, what can we then expect of a member of Congress ? 105 XYSTROM'S CALCULATOR. This calculating machine consists of a silvered brass plate of about nine inches in diameter, on which are fixed two movable arms, extending from the centre to the periphery. On the plate are engraved a number of curved lines in such form and divisions that with their intersection with the arms, the most complicated calculations can be performed almost instantly. The arrangement for trigonometrical calculations is such that it is not necessary to notice the functions sine^ cosine^ tangent^ &c., operating only by the angle expressed in degress and minutes, and without any tables, which makes it so easy that any one who can read figures, will be able to solve trigonometrical questions. Any kind of calculation can be performed on this instrument, no matter how complicated it may be, whilst there is nothing intricate in its use. The aiithor, who is the inventor of the calculator, has thoroughly tested its practical utility. All the calcu- lations in Nystrom's Pocket Book of Mechanics and Engineering have been computed by this instrument. 106 Teachers are generally dependent upon text book for examples, when it is easy for the pupil, knowing where it comes from, to be furnished with an answer; but with the calculator, the teacher can vary the examples ad Uhitum, and the answer is almost instantly at hand, while the pupil is thrown on his own resources for the proper solution, and his real acquire- ment is tested. The price of the Calculator, with complete descrip- tion and examples how to use it, $20. Manufactured by Wm. J. Young, 43 North Seventh Street. Sold by James W. Queen «fe Co., 924 Chestnut Street, Philadelphia. NYSTROM'S PUBLISHED BY J. B. LIF»P»I]SrCOTT & CO. PHILADELPHIA. TRUBNEE k CO., LONDON. This Pocket Book is now in its fifth edition, revised and enlarged with fifty new pages, and contains new and original matter not to be found in other Engineer- ing books. Price, §1.50. v*n.t'£r*tua^>->> >>it*.i: im ■^ UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. •4^^^ k MAY 231972^5 JUN 1 6 1972 EC'DLD MAY 3 RECEi FEB 2 2 1983 CJRCULATJON DEPtL 72 -4 PM 3 % LD 21-100m-9,'47(A5702sl6)476 ^1 m Si5l0946 A/? U. C. BERKELEY LIBRARIES CDbl3^fl53S i B \ ;;3- 1 ■ |ii M