Chapter IX: MISCELLANEOUS.--Spanish Fisheries.--Noticeable objects (4)

_Carpinus orientalis_[2] (the Oriental hornbeam) principally differs from our native species in its smaller size, the lesser leaves with downy petioles, and the green, much-lacerated bractlets. It is a native of the south of Europe, whence it extends to the Caucasus, and probably also to China; the Carpinus Turczaninovi of Hance scarcely seems to differ, in any material point at any rate, from western examples of C. orientalis. According to Loudon, it was introduced to this country by Philip Miller in 1739, and there is no doubt that it is far from common even now. It is, however, well worth growing; the short twiggy branches, densely clothed with dark green leaves, form a thoroughly efficient screen. The plant bears cutting quite as well as the common hornbeam, and wherever the latter will grow this will also succeed. In that very interesting compilation, "Hortus Collinsonianus," the following memorandum occurs: "The Eastern hornbeam was raised from seed sent me from Persia, procured by Dr. Mounsey, physician to the Czarina. Received it August 2, 1751, and sowed it directly; next year (1752) the hornbeam came up, which was the original of all in England. Mr. Gordon soon increased it, and so it came into the gardens of the curious. At the same time, from the same source, were raised a new acacia, a quince, and a bermudiana, the former very different from any in our gardens." This memorandum was probably written from recollection long afterward, with an error in the dates, and the species was first entered in the catalogue as follows: "Azad, arbor persica carpinus folio, Persian hornbeam, raised from seed, anno 1747; not in England before." It appears, however, from Rand's "Index" that there was a plant of it in the Chelsea Garden in 1739. The name duinensis was given by Scopoli, because of his having first found it wild at Duino. As, however, Miller had previously described it under the name orientalis, that one is adopted in accordance with the rule of priority, by which must be decided all such questions in nomenclature.

[Footnote 2: IDENTIFICATION.--Carpinus orientalis. Miller,
"Gardener's Dictionary," ed. 6 1771; La Marck, Dict, i., 107;
Watson, "Dendrologia Britannica," ii., tab. 98; Reich. Ic. fl.
Germ. et Helvet., xxii., fig, 1298; Tenore, "Flora Neapolitana,"
v., 264; Loudon, Arb. et Fruticet. Brit., iii., 2014, Encycl.
Trees and Shrubs, p. 918; Koch, "Dendrologie." zweit, theil zweit,
abtheil, p. 4. C. duinensis, Scopoli, "Flora Carniolica," 2 ed.,
ii., 243, tab. 60; Bertoloni, "Flora Italica," x., 233; Alph. De
Candolle in Prodr., xvi. (ii.), 126.]

_The American Hornbeam_ [3] also known under the names of blue beech, water beech, and iron wood, although a less tree than our native species, which it resembles a good deal in size of foliage and general aspect, is nevertheless a most desirable one for the park or pleasure ground, on account of the gorgeous tint assumed by the decaying leaves in autumn. Emerson, in his "Trees and Shrubs of Massachusetts," pays a just tribute to this tree from a decorative standpoint. He says: "The crimson, scarlet, and orange of its autumnal colors, mingling into a rich purplish red, as seen at a distance, make it rank in splendor almost with the tupelo and the scarlet oak. It is easily cultivated, and should have a corner in every collection of trees." It has pointed, ovate oblong, sharply double serrate, nearly smooth leaves. The acute bractlets are three-lobed, halberd-shaped, sparingly cut-toothed on one side. Professor C.S. Sargent, in his catalogue of the "Forest Trees-of North America," gives the distribution, etc., of the American hornbeam as follows: "Northern Nova Scotia and New Brunswick, through the valley of St. Lawrence and Lower Ottawa Rivers, along the northern shores of Lake Huron to Northern Wisconsin and Minnesota; south to Florida and Eastern Texas. Wood resembling that of ostrya (hop hornbeam). At the north generally a shrub or small tree, but becoming, in the Southern Alleghany Mountains, a tree sometimes 50 feet in height, with a trunk 2 feet to 3 feet in diameter." It will almost grow in any soil or exposition in this country.

[Footnote 3: IDENTIFICATION.--Carpinius caroliniana, Walter,
"Flora Caroliniana," 236; C. americana, Michx. fl. bor. Amer.,
ii., 201; Mich. f. Hist. des. Arbres Forestiers de l'Amerique
Septentrionale, iii., 57, tab. 8; Watson, "Dendrologia
Britannica," ii., 157; Gray, "Manual of the Botany of the Northern
United States," p. 457.]

_Carpinus viminea_[4] is a rather striking species with long-pointed leaves; the accompanying figure scarcely gives a sufficiently clear representation of their long, tail-like prolongations. Judging from the height at which it grows, it would probably prove hardy in this country, and, if so, the distinct aspect and graceful habit of the tree would render it a decided acquisition. It is a moderate-sized tree, with thin gray bark, and slender, drooping warted branches. The blade of the smooth leave measures from 3 inches to 4 inches in length, the hairy leaf-stalk being about half an inch long. It is a native of Himalaya, where it occurs at elevations of from 5000 to 7000 feet above sea-level. As in our common hornbeam, the male catkins appear before the leaves, and the female flowers develop in spring at the same time as the leaves. The hard, yellowish white wood--a cubic foot of which weighs 50 lb.--is used for ordinary building purposes by the natives of Nepaul.

[Footnote 4: IDENTIFICATION.--Carpinus viminea, Lindl. in Wall.
Plant. Asiat. Rar., ii., p. 4, t. 106; D.C. Prodr., xvi., ii.,
127. Loudon, "Arboretum et Fruticetum Britannicum," iii., p. 2014;
Encycl. of Trees and Shrubs, p. 919. Brandis, "Forest Flora,"
492.]

GEORGE NICHOLSON. Royal Gardens, Kew.

* * * * *

FRUIT OF CAMELLIA JAPONICA.

The fruiting of the camellia in this country being rather uncommon, we have taken the opportunity of illustrating one of three sent to us a fortnight ago by Mr. J. Menzies, South Lytchett, who says: "The fruits are from a large plant of the single red, grown out of doors against a wall with an east aspect, and protected by a glazed coping 4 feet wide. The double, semi-double, and single varieties have from time to time borne fruit out of doors here, from which I have raised seedlings, but have hitherto failed to get any variety worth sending out or naming."

In the annexed woodcut the fruit is represented natural size. Its appearance is somewhat singular. It is very hard, and has a glazed appearance like that of porcelain. The color is pale green, except on the exposed side, which is dull red. It is furrowed like a tomato, and on the day after we received it the furrows opened and exposed three or four large mahogany-brown seeds embedded in hard pulp.--_The Garden._

* * * * *

[SCIENCE.]

A NEW RULE FOR DIVISION IN ARITHMETIC.

The ordinary process of long division is rather difficult, owing to the necessity of guessing at the successive figures which form the divisor. In case the repeating decimal expressing the _exact_ quotient is required, the following method will be found convenient:

_Rule for division_.

_First._ Treat the divisor as follows:

If its last figure is a 0, strike this off, and treat what is left
as the divisor.

If its last figure is a 5, multiply the whole by 2, and treat the
product as the divisor.

If its last figure is an even number, multiply the whole by 5, and
treat the product as a divisor.

Repeat this treatment until these precepts cease to be applicable. Call the result the _prepared divisor_.

_Second._ From the prepared divisor cut off the last figure: and, if this be a 9, change it to a 1, or if it be a 1, change it to a 9; otherwise keep it unchanged. Call this figure the _extraneous multiplier_.

Multiply the extraneous multiplier into the divisor thus truncated, and increase the product by 1, unless the extraneous multiplier be 7, when increase the product by 5. Call the result the _current multiplier_.

_Third._ Multiply together the extraneous multiplier and all the multipliers used in the process of obtaining the prepared divisor. Use the product to multiply the dividend, calling the result the _prepared dividend_.

_Fourth._ From the prepared dividend cut off the last figure, multiply this by the current multiplier, and add the product to the truncated dividend. Call the sum the _modified dividend_, and treat this in the same way. Continue this process until a modified dividend is reached which equals the original prepared dividend or some previous modified dividend; so that, were the process continued, the same figures would recur.

_Fifth._ Consider the series of last figures which have been successively cut off from the prepared dividend and from the modified dividends as constituting a number, the figure first cut off being in the units' place, the next in the tens' place, and so on. Call this the _first infinite number_, because its left-hand portion consists of a series of figures repeating itself indefinitely toward the left. Imagine another infinite number, identical with the first in the repeating part of the latter, but differing from this in that the same series is repeated uninterruptedly and indefinitely toward the right into the decimal places.

Subtract the first infinite number from the second, and shift the decimal point as many places to the left as there were zeros dropped in the process of obtaining the prepared divisor.

The result is the quotient sought.

_Examples._

1. The following is taken at random. Divide 1883 by 365.

_First._ The divisor, since it ends in 5, must be multiplied by 2, giving 730. Dropping the O, we have 73 for the prepared divisor.

_Second._ The last figure of the prepared divisor being 3, this is the extraneous multiplier. Multiplying the truncated divisor, 7, by the extraneous multiplier, 3, and adding 1, we have 22 for the current multiplier.

_Third._ The dividend, 1883, has now to be multiplied by the product of 3, the extraneous multiplier, and 2, the multiplier used in preparing the divisor. The product, 11298, is the prepared dividend.

_Fourth._ From the prepared dividend, 11298, we cut off the last figure 8, and multiply this by the current multiplier, 22. The product, 176, is added to the truncated dividend, 1129, and gives 1305 for the first modified divisor. The whole operation is shown thus:

1 8 8 3
6
-------
1 1 2 9|8
1 7 6 -
-----
1 3 0|5
1 1 0 -
-----
2|4 0
8 8 ---
---
|9 0
-----
1 9|8
1 7 6 -
-----
1 9|5
1 1 0 -
-----
1 2|9
1 9 8 -
-----
2|1 0
2 2 ---
2 4

We stop at this point because 24 was a previous modified dividend, written under the form 240 above. Our two infinite numbers (which need not in practice be written down) are, with their difference:

. .
10,958,904,058 . .
10,958,904,109.5890410958904
----------------------------
. .
51.5890410958904
. .
Hence the quotient sought is 5.158904109.

_Example 2._ Find the reciprocal of 333667.

The whole work is here given:

3 3 3 6 6|7 |7
2 3 3 5 6 7 - 1 6 3 4 9 6|9
2 1 0 2 1 0 3 -
-------------
2 2 6 5 5 9|9
2 1 0 2 1 0 3 -
-------------
2 3 2 8 6 6|2
4 6 7 1 3 4 -
-----------
7 0 0 0 0 0

. .
_Answer_, 0.000002997.

_Example 3._ Find the reciprocal of 41.

_Solution._--

4|1 |9
----- -----
3 7|9 3 3|3
- 1 1 1 -
-----
1 4|4
1 4 8 -
-----
1 6|2
7 4 -
---
9 0
. .
_Answer_, 0.02439.

C.S. PEIRCE.

* * * * *

[SCIENCE.]

EXPERIMENTS IN BINARY ARITHMETIC.

Those who can perform in that most necessary of all mathematical operations, simple addition, any great number of successive examples or any single extensive example without consciousness of a severe mental strain, followed by corresponding mental fatigue, are exceptions to a general rule. These troubles are due to the quantity and complexity of the matter with which the mind has to be occupied at the same time that the figures are recognized. The sums of pairs of numbers from zero up to nine form fifty-five distinct propositions that must be borne in memory, and the "carrying" is a further complication. The strain and consequent weariness are not only felt, but seen, in the mistakes in addition that they cause. They are, in great part, the tax exacted of us by our decimal system of arithmetic. Were only quantities of the same value, in any one column, to be added, our memory would be burdened with nothing more than the succession of numbers in simple counting, or that of multiples of two, three, or four, if the counting is by groups.

It is easy to prove that the most economical way of reducing addition to counting similar quantities is by the binary arithmetic of Leibnitz, which appears in an altered dress, with most of the zero signs suppressed, in the example below. Opposite each number in the usual figures is here set the same according to a scheme in which the signs of powers of two repeat themselves in periods of four; a very small circle, like a degree mark, being used to express any fourth power in the series; a long loop, like a narrow 0, any square not a fourth power; a curve upward and to the right, like a phonographic _l_, any double fourth power; and a curve to the right and downward, like a phonographic _r_, any half of a fourth power; with a vertical bar to denote the absence of three successive powers not fourth powers. Thus the equivalent for one million, shown in the example slightly below the middle, is 2^{16} (represented by a degree-mark in the fifth row of these marks, counting from the right) plus 2^{17} + 2^{9} (two _l_-curves in the fifth and third places of _l_-curves) plus 2^{18} + 2^{14} + 2^{6} (three loops) plus 2^{19} (the _r_-curve at the extreme left); while the absence of 2^{3}, 2^{2}, and 2^{1} is shown by the vertical stroke at the right. This equivalent expression may be verified, if desired, either by adding the designated powers of two from 524,288 down to 64, or by successive multiplications by two, adding one when necessary. The form of characters here exhibited was thought to be the best of nearly three hundred that were devised and considered and in about sixty cases tested for economic value by actual additions.

In order to add them, the object for which these forty numbers are here presented in two notations, it is not necessary to know just _why_ the figures on the right are equal to those on the left, or to know anything more than the order in which the different forms are to be taken, and the fact that any one has twice the value of one in the column next succeeding it on the right. The addition may be made from the printed page, first covering over the answer with a paper held fast by a weight, to have a place for the figures of the new answer as successively obtained. The fingers will be found a great assistance, especially if one of each hand be used, to point off similar marks in twos, or threes, or fours--as many together as can be certainly comprehended in a glance of the eye. Counting by fours, if it can be done safely, is preferable because most rapid. The eye can catch the marks for even powers more easily in going up and those for odd powers (the _l_ and _r_ curves) in going down the columns. Beginning at the lower right hand corner, we count the right hand column of small circles, or degree marks, upward; they are twenty-three in number. Half of twenty-three is eleven and one over; one of these marks has therefore to be entered as part of the answer, and eleven carried to the next column, the first one of _l_-curves. But since the curves are most advantageously added downward, it is best, when the first column is finished, simply to remember the remainder from it, and not to set down anything until the bottom is reached in the addition of the second column, when the remainders, if any, from both columns can be set down together. In this case, starting with the eleven carried and counting the number of the _l_-curves, we find ourselves at the bottom with twenty-four--twelve to carry, and nothing to set down except the degree mark from the first column. With the twelve we go up the adjoining loop column, and the sum must be even, as this place is vacant in the answer; the _r_-curve column next, downward, and then another row of degree marks. The succession must be obvious by this time. When the last column, the one in loops to the extreme left, is added, the sum has to be reduced to unity by successive halvings. Here we seem to have eleven; hence we enter one loop, and carry five to the next place, which, it must be remembered, is of _r_-curves. Halving five we express the remainder by entering one of these curves, and carry the quotient, two, to the degree mark place. Halving again gives one in the next place, that of _l_-curves; and the work is complete.

It is recommended that this work be gone over several times for practice, until the appearance and order of the characters and the details of the method become familiar; that, when the work can be done mechanically and without hesitation, the time occupied in a complete addition of the example, and the mistakes made in it, be carefully noted; that this be done several times, with an interval of some days between the trials, and the result of each trial kept separate; that the time and mistakes by the ordinary figures in the same example, in several trials, be observed for comparison. Please pay particular attention to the difference in the kind of work required by the two methods in its bearing on two questions--which of them would be easier to work by for hours together, supposing both equally well learned? and in which of them could a reasonable degree of skill be more readily acquired by a beginner? The answer to these questions, if the comparison be a fair one, is as little to be doubted as is their high importance.

_Example in addition by two notations_

77,823,876
14,348,907
8,654,912
5,764,801
4,635,857
1,594,323
6,417,728
4,782,969
83,886,075
34,012,224
2,903,111
48,828,125
1,724,826
7,529,536
43,344,817
10,000,000
8,334,712
1,953,125
11,308,417
759,375
21,180,840
9,765,625
18,643,788
1,000,000
44,739,243
1,889,568
2,517,471
40,353,607
4,438,414
1,679,616
23,708,715
11,890,625
945,754
823,543
15,308,805
60,466,176
30,685,377
10,077,696
19,416,381
43,046,721
===========
740,685,681

Eight volunteer observers to whom this example has already been submitted showed wide difference in arithmetical skill. One of them took but a few seconds over two minutes, in the best of six trials, to add by the usual figures, and set down the sum, but one figure in all the six additions being wrong; another added once in ten minutes fifty-seven seconds, and once in eleven minutes seven seconds, with half the figures wrong each time. The last-mentioned observer had had very little training in arithmetical work, but perhaps that gave a fairer comparison. In the binary figures she made three additions in between seven and eight minutes, with but one place wrong in the three. With four of the observers the binary notation required nearly double the time. These observers were all well practiced in computation. Their best record, five minutes eighteen seconds, was made by one whose best record was two minutes forty seconds in ordinary figures. The author's own best results were two minutes thirty-eight seconds binary, and three minutes twenty-three seconds usual. He thus proved himself inferior to the last observer, as an adder, by a system in which both were equally well trained; but a greater familiarity (extending over a few weeks instead of a few hours) with methods in binary addition enabled him to work twice as fast with them. Of the author's nine additions by the usual figures, four were wrong in one figure each; of his thirty-two additions by different forms of binary notation, five were wrong, one of them in two places. One observer found that he required one minute thirty-three seconds to add a single column (average of five tried) by the usual figures, and fifteen seconds to count the characters in one (average of six tried) by the binary. Though these additions were rather slow, the results are interesting. They show, making allowance for the greater number of columns (three and a third times as many) required by the binary plan, a saving of nearly half; but they also illustrate the necessity of practice. This observer succeeded with the binary arithmetic by avoiding the sources of delay that particularly embarrass the beginner, by contenting himself with counting only, and not stopping to divide by two, to set down an unfamiliar character, or to recognize the mark by which he must distinguish his next column. One well-known member of the Washington Philosophical Society and of the American Association for the Advancement of Science, who declined the actual trial as too severe a task, estimated his probable time with ordinary figures at twenty minutes, with strong chances of a wrong result, after all.

These statistics prove the existence of a class of persons who can do faster and more reliable work by the binary reckoning. But too much should not be made of them. Let them serve as specimens of facts of which a great many more are to be desired, bearing on a question of grave importance. Is it not worth our while to know, if we can, by impartial tests, whether the tax imposed on our working brains by the system of arithmetic in daily use is the necessary price of a blessing enjoyed, or an oppression? If the strain produced by greater complexity and intensity of mental labor is compensated by a correspondingly greater rapidity in dealing with figures, the former may be the case. If, on the contrary, a little practice suffices to turn the balance of rapidity, for all but a small body of highly drilled experts, in favor of an easier system, the latter must be. This is the question that the readers of _Science_ are invited to help in deciding. The difficulties attending a complete revolution in the prevalent system of reckoning are confessedly stupendous; but they do not render undesirable the knowledge that experiment alone can give, whether or not the cost of that system is unreasonably high; nor should they prevent those who accord them the fullest recognition from assisting to furnish the necessary facts.

Those who are willing to undertake the addition on the plan proposed or on any better plan, or who will submit it to such acquaintances, skilled or unskilled, as may be persuaded to take the trouble to learn the mechanism of binary adding, will confer a great favor by informing the writer of the time occupied, and number of mistakes made, in each addition. All observations and suggestions relating to the subject will be most gratefully received.

Henry Farquhar.

Office of U.S. Coast Survey, Washington, D.C.

* * * * *

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