Continuatio argumenti de mensura sortis ad fortuitam successionem rerum naturaliter contingentium applicata

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1 Continuatio argumenti de mensura sortis ad fortuitam successionem rerum naturaliter contingentium applicata Daniel Bernoulli Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae Vol. XV p (1771) 1. In our previous studies concerning that argument we have examined the hypothesis thus far having the appearance of truth in the first place to consider, that the falsehood of it will have commenced to come in the midst of suspicion finally after countless experiments; I understand the equal proclivity of nature to forming each sex. Now certainly all experts admit with one voice, nature to favor the male sex more than the other or at least to this point to have favored more always. Certainly can it e that it has happened y lind chance or y the command of a natural law? For my part the prior is possile, the other certainly y far most having the appearance of truth and most proale; let us disregard the words and let us ponder the thing itself. Thus the value of the work will e as of the individual events which are ale to happen, let us examine the proaility of the cases for this other hypothesis, with respect to which nature is more fecund in forming a male descendent, than in the other and it in any given ratio ut the same uniformly which I shall name a to. I have encountered a new question, more amplified than the first infinities, esides the expectation concisely expressed with an elegant and sufficiently simple formula, which now I shall explain. 2. Again let there e, just as in the second paragraph of the preceding dissertation, the numer of annual irths = 2N and, in order that we may indicate the thing in a mathematical term, let us put for any irth the sex to e determined in this manner, that into an urn tickets have een restored some lack for the male sex others white for the next sex to e specified; the numer of lack tickets will e = a, of white tickets = : moreover the extracted ticket will announce the sex of any irth next restoring the ticket into the urn; ecause if y this manner the sex for 2N irths is determined, it is sought how much e the proaility that the numer of oys ecome = m precisely and the numer of girls = 2N m exactly. Now the theory of cominations will give, if only all of the disposed will have een arranged, the following more general formula, Continuation of the argument concerning the measurement of chance applied to the random succession of natural events. Translated y Richard J. Pulskamp, Department of Mathematics and Computer Science, Xavier University, Cincinnati OH. 1

2 which certainly sets up the value of the sought proaility: 2N.(2N 1).(2N 2).(2n 3)... (2N m + 1) ) ( ) m 2N m.2 2N a + 3. I have marveled at the simplicity of the manner, y which this more general theory emraces the other sent ahead y us on ehalf of each sex having equal power: for with a = put immediately it is oserved to happen ) ( ) m 2N 2 = 1 a + and exactly to produce the same formula plainly, as we have set forth in the second paragraph of the first dissertation. But if a very small inequality intervenes etween a and, immediately thenceforth a conspicuous difference will arise etween the proailities computed under each hypothesis, as often as the numers for N are assumed larger; namely oth proailities for the same numers m and N are as unity to the numer ) ( ) m 2N 2, a + which ratio generally recedes to a ratio of equality completely as the magnitude of the numer m and this quality makes availale a criterion to us not at all to e despised in distinguishing a law of nature, if natality tales, for many years, are at hand. Look at an example of this thing. The numer of all irths will e= or N = and let the discussion e of the special case, y which that sum of irths from each sex is divided into two parts perfectly equal to each other: we will have m = 10000; let e put a/ = 1.055, which value corresponds to oservations not adly: thus it ecomes ) ( ) m 2N 2 = a + ( ) ( ) = Therefore the proaility, which for this case is however a very small thousandth, will e one thousand two hundred ninety six greater if only there will have een a/ = 1 than if there is a/ = 1.055; ut we have seen in our preceding dissertation in 1 the seventh paragraph the proaility for the first position to e ; therefore the or proaility for the second position will e = 1. If we add all to this very small proaility, in which the numer m is suppressed elow the numer N, however much the prior is ale to ecome y the methods to ten thousand as the numer m vanishes completely, yet the sum of all these proailities for ten thousand cases is not stronger than the proaility is for a single case, where it is assumed m = N; whence it is deduced, if the question will have een how much the proaility e that at London more girls may e orn than little oys within a year or in equal numer 1 at least, this proaility to e less than ; ut it is certainly similar, that it may happen just aout once in any course of years. A tale exists here and there, where annually from the year 1664 to the year 1758 the numer is indicated so many 2

3 of little girls as of aptized little oys at London in the Episcopal Church, where it is to see never within 95 years to have happened that the numer of girls was equal, still less greater, than the numer of oys, although the numer of all annual aptisms notaly was less than and truly it should have een ale to happen much more easily; in the year 1703 the girls have approached most to equality with the oys; certainly 7683 girls have een aptized and 7765 little oys; ut of course this small difference has een most difficult to overcome y far. 4. The formula exposed at the end of the second paragraph exhiits the nature of our argument excellently. Under the first hypothesis, where is put a =, the proaility decreases from the middle toward one of the two extremities; under the second hypothesis, where is put a >, first it increases to a certain term eyond which it decreases; near the middle, where m = N, the proaility under the first hypothesis exceeds greatly the proaility under the other hypothesis; ecause certainly in the former it decreases in the latter it increases, the place will e where the same proaility is under each hypothesis, the other place where the proaility, under the latter hypothesis, may e doule, triple, quadruple &c. now these places or values I shall define m; ut it is required, that the factor ) ( ) m 2N 2 a + e put successively equal to 1, 2, 3, 4 &c. and thence the numer m is determined. Let us egin with the first equation and we will find m = 2N(log(a + ) log(2)). log a log Let us call this first value A and thus we will have successively Thus generally there will e m = A log 2 m = A + log a log log 3 m = A + log a log log 4 m = A + log a log m = A + log φ log a log if it should e desired that the two proailities must themselves e as 1 : φ. Let us descend to a numerical example. 5. Let there e further N = and a = 1055/1000; there will e A = and generally m = log φ log

4 or, with the common logarithm employed, whence if there is put successively m = log φ φ = 1 there will e had m = φ = 2 m = φ = 3 m = φ = 4 m = φ = 5 m = φ = 10 m = It is evident from this here how the ratio which intercedes etween the proailities for the two positions a = 1 and a = increases immensely. It is understood likewise with respect to which how often the numer of oys orn increased y forty-three is located so often is the ratio etween the two proportional proailities multiplied tenfold. 6. The relation etween φ and m relates to the logarithm so that y the most simple operation the numer m is ale to e indicated, for which the ratio φ may otain the given value. Let, for the sake of an example, for the numers assumed in the preceding paragraph, m indicating the numer of oys, which happens ten times one thousand y a thousand thousand times more easily, specify a = than specify a = 1. In this example let φ = 10, 000, 000, 000 and log φ = 10 therefore ( 5) m = = y which one must not e amazed y the almost extraordinary distinction of the proailities for the case, y which the numer of oys crosses the center, y the small numer 564 among But if therefore with the rarest case it will have happened that among irths little oys had een counted and exactly 9436 girls, who, understanding of these things, will estalish the nature according to each sex the proclivity to e formed asolutely equal? It is certain anyhow, a case of this kind to ecome more proale y 10, 000, 000, 000 times, if there will have een a = 1.055, ever so much also then certainly scarcely it merits to restore 1 among the possiles, seeing that the proaility of that case alone is only = If further therefore that least proaility is increased, in order that it comprehends all cases, in which the numer of little oys crosses the numer 10564, scarcely thence it will ecome ten times greater, so much I am ale to decide without the estalished calculus; thus each proaility will ecome only = , y which neglected it will e permitted to affirm under no circumstances aout to e that the numer of oys orn may ascend to out of irths, even if to the male sex its prerogative may e assigned, as the value a = indicates. 7. We have seen presently, what very little appearance of the truth there is, that for irths the numer of oys ever may ascend to and of girls truly e kept down to 9436 and thus the difference etween oth sexes may spread to 1128, even if nature may favor the male sex over the other in ratio 1055 to Thus curious of this thing I have consulted the London tale cited aove, which the celerated Süssmilch reckons in the second part of his distinguished work, 1 to which at the end many tales 1 J. P. Süssmilch, Die Göttliche Ordnung in den Veränderungen des menschlichen Geschlechts, aus der Geurt, Tod, und Fortpflanzung desselen erwiesen, Berlin 1741 (1 st edition). 4

5 of this kind have een added: I have examined years: where the numer of oys most surpassed the girls; the remarkale ones to me have een seen the year 1676, where 6552 little oys orn or rather aptized and 5847 little girls are indicated, next the year 1698, where 8426 males and 7627 females; finally the year 1717, where 9630 males and 8845 of the other sex are indicated. But I have shown the differences etween each sex to e changed in suduplicated ratio of the numers 2N as the same proaility is returned: therefore y this correction employed I have discovered none out of the three alleged years to e, which are worthy of greater attention, than if for irths the excess of oys over girls will have een just aout 900, which excess wants much yet from Nevertheless I shall not leave unsaid the especially rare year 1749 in view of all the others, where namely aptized 7288 little oys and so many 6172 little girls are indicated; we have here the excess of oys = 1116, while the sum of all irths was at least = 13460; therefore the aforesaid excess 1116 must e increased in suduplicate ratio of the numers to 20000, y that reduction accomplished the aforesaid excess is changed to 1361; now certainly that excess notaly surpasses the excess 1128, which not without reason we have supposed in the very long series of more than thousands of years is going to happen once with difficulty; therefore I persuade myself an error to have crept into one or the other of the numers 6172 and 7288; without dout much more easily it is, that in tales stuffed with so many numers and most often copied an error may creep in sometime or other than that an extraordinary inequality may find a place; on the other hand I suppose in the place of 6172 little daughters to have een written 6972; certainly with this change accomplished the relation of one numer etween either sex ecomes especially proale, which had een not so much impossile. I have examined the error suspected of the numers, which indicates the sum of the aptized young daughters within the ten years from the year 1741 to the end of the year 1750; in the tale for the sum is put, which used with my correction agrees perfectly to the thing itself; therefore the error has een committed either y the writer or y the typesetter, nor do I dout in fact the annals of London are going to confirm my conjecture. It is permitted to add a word concerning the aptism tale, which the same author produces page 13 for the capital city of Austria; a single inspection of it for me has provoked anger; it contains nothing, not even I fear to say, except pure fictions, written down with wandering pen, how deceitful is this tale, without our calculations it is scarcely understood nor am I amazed, ecause the celerated Süssmilch must have considered it worthy to insert into his extraordinary work, producing the reported has restored unrestrained confidence to each one. 8. It is understood from the previous, that with the assumed numer m > 2N(log(a + ) log(2) log a log the proaility, under the hypothesis a >, may surpass excessively the proaility under the hypothesis a = ; the contrary is otained when it is assumed m < 2N(log(a + ) log(2). log a log 5

6 Of course, with the significance of the letter φ retained, now the ratio etween each proaility will e expressed y 1 φ and since there is there will e had ( 4) log 1 = log φ, φ m = A log φ log a log ; and, for example m = 10000, there ecomes ( 5) m = log φ, where now φ designates, how much the proaility, under the hypothesis a >, must e surpassed y the proaility under the hypothesis a =. Let e put again φ = 10, 000, 000, 000 and there ecomes m = or m = Thus understand this thing. There is sought, under the hypothesis a =, the proaility that the numer of oys is = and that proaility will e discovered just aout = or = ; 10,000,000,000, 1 if it is assumed of this least small fraction it will e held for the hypothesis a = the proaility that the numer of oys is = Smallness of this kind is hardly understood y the mind. Finally if all and individual proailities are united into a sum that the numer of oys is suppressed elow 9704 and there is provided y this summation the proaility to ecome ten times greater, the comined proaility ecomes = 1 1,416,000,000,000,000, y which disregarded it is permitted to affirm, it not e ale to happen that the numer of oys cross the limits and 9704, if it will have een a = nor the numer of girls the limits 9436 and Now I am undertaking another question, y what then is the numer m the annual male offspring given with greatest proaility in view of all the rest? Truly, under the hypothesis a =, in the first paper I have assumed without demonstration, ecause at that time the thing is clear y itself, to e made m = N. But when an inequality etween a and is supposed, the present question assumes another form. We will recur to the formula exposed at the end of the second paragraph, which for any numer m expresses its proaility, certainly 2N.(2N 1).(2N 2).(2n 3)... (2N m + 1) ) ( ) m 2N m.2 2N a + In that formula the value certainly of the indefinite factor egins to decrease immediately and moreover the numer m is assumed > N; ut in fact certainly the other variale factor ( ) a m seeing that it proceeds to grow continuously, it is evident to e ale to have a position, where the product is a maximum out of the two factors; hence in some way or other as if eccentricity. It will e permitted to define that place of maximum proaility from it certainly, ecause the proaility must e the same for the 6

7 two indices nearest to m and m + 1. Moreover with the assumed 2N.(2N 1).(2N 2).(2n 3)... (2N m + 1) m.2 2N = S the proaility for the index m is ) ( ) m 2N 2 = S a + and equally for the index m + 1 the proaility emerges = 2N m m + 1 S ) ( ) m+1 2N 2. a + Therefore with an equation made etween the two proailities, it is discovered or 2N m m + 1 a = 1 m = a +, ecause certainly the term is as if incomparaly greater than, it will e ale simply to put m = a + and the numer of girls or 2N m = 2N a + so that the two numers themselves are in the ratio a to, that which confirms our formulas exceptionally. 10. Thus therefore in our example, where we have put 2N = 20000, the maximum proaility falls at the numer m = 10268; eyond and efore this location the proaility decreases, from the eginning at least most slowly, then more quickly, next enormously. We shall note here in passing what that point, concerning which 4 and 5 we have asserted, where the proaility is the same under each hypothesis, it is in the position at the center etween the two points of maximum proaility under each hypothesis; it is namely for this point m = 10134, which numer is the center etween and and this property generally takes place, if the difference etween a and is small. 11. It serves chiefly to consider the aforementioned position, which makes m = a + and for which the maximum proaility arises, just as if a fixed point and thus all calculations to assume that the distance from this point is considered just as if from the center of strength: which in fact have een said they not yet support the computation 7

8 enough: therefore the work is given that for whatever given index m the proaility may e determined from some definite formula, in all events as near as possile since it cannot happen in all rigor. I have undertaken this path in the first paper certainly not without success; where concerning the thing in the first place the fifth paragraph appears; next the eighteenth. 12. Let e assumed, for the sake of revity, a+ = M, thus so that M denotes the numer of oys given with maximum proaility and let the index e assumed m = M + µ where µ will record the excess of the index aove the numer corresponding to the maximum proaility: let the proaility e assumed in addition, for the index M + µ = π, which true value now we have indicated aove in the second and ninth paragraph ut expressed y an indefinite formula, incomputale for large numers. The mind considers to examine again, can it e that to this indefinite formula another nearest the actual and definite is ale to e sustituted; we will consider the quantities µ and π just as if coordinate variales. But it is clear from the indefinite formula itself the proaility to e, for the index nearest M + µ + 1, or = 2N m m + 1 a π 2N m µ M + µ + 1 a π, which if it is sutracted from the preceding proaility π the difference of one or the other proaility will e = π 2N M µ M + µ + 1 a π. For certainly a second time I will suppose this difference to e compared with the difference of the two nearest indices, that is, compared with unity, just as dπ to dµ, which certainly without any sensile error is ale to e supposed on account of the proximate relationship of the two indices; moreover this supposition furnishes the following equation dπ ( π = 1 2N M µ M + µ + 1 a ) dµ. In that equation for the quantity M I shall restore the value of it a +, that y so much more the quantities, which in the end of the calculation are ale to e neglected, are ale to e distinguished from one another; that fact ecomes y restatement dπ ( ) π = : (a + ) µa : ) 1 dµ 2na : (a + ) + µ + 1 8

9 or or finally dπ π = µ µa : : (a + ) + µ + 1 dµ dπ π = µ µa : M + µ + 1 dµ. 13. Thus the aforementioned equation is integrated so that with µ = 0 assumed it happens π = Q, where y Q I understand the maximum proaility, which has location as index M or a+. Thus it produces log Q π = a + µ a + (M + 1) log M + µ + 1 M log M + µ + 1. M + 1 Because certainly this equation must e serviceale only to the examples to e computed, in which the numer µ is very much less than the numer M, seeing that in the others the proaility nearly vanishes and is not deserved that a ratio is considered of it, out of the thing there will e in the next to last term the quantity log M + µ + 1 M + 1 to convert into series; in this series it will suffice to have considered the first three terms and thus to assume log M + µ + 1 = µ M + 1 M ( ) 2 µ + 1 ( ) 3 µ ; 2 M M + 1 in this way thus the equation will e ale to e set log Q π = a + ( µµ a 2(M + 1) µ 3 3(M + 1) 2 + log M + µ + 1 ) M + 1 or with c assumed for the numer of which the hyperolic logarithm is unity or Q π = M + µ + 1 M + 1 π Q = M + 1 M + µ + 1 c [ c a+ [ a+ ] µµ 2(M+1) µ3 3(M+1) 2 ] µµ 2(M+1) µ3 3(M+1) 2 Because if we wish to relax more from scrupulosity, it will e permitted with that simple formula, π Q = a+ c 2 µµ M Certainly this last formula is perfectly the same with that, which I have exhiited in the first paper 18 and which I have confirmed in the following very small tale; with on the other hand a = assumed, it happens likewise M = N and thus there appears simply π Q = 1 N 9 c µµ

10 14. The aforementioned equation will e so much more accurate, as the difference etween a and is supposed less and as the numer µ is likewise less, ecause each is appropriate enough to our principle; therefore it will e worthwhile to expose this equation to last examination. Let the value indicated for the letter M of the one in the twelfth paragraph e restored, of course thus the exponent will e a + 2 µµ M = (a + )2 4a µµ N ; a+ ; let = 1 and a = 1 + α e assumed, where α is put much smaller than unity; there will e had (a + ) α + αα = = 1 + αα 4a 4 + 4α 4 + 4α ; here clearly the term added to unity is ale to e neglected and indeed to e assumed (a+) 2 4a = 1, y which fact the exponent ecomes a + 2 and thus there is ale simply to e put µµ M = µµ N π Q = µµ c N or = 1. c µµ N Thus therefore, for each value a, the proaility is expressed without change in the same manner, ut, from the location of the distance of the term from the middle, the distance from the given term with the maximum proaility is understood through µ, which special character certainly merits each attention. 15. And also the proaility itself of the term, which is a maximum, with the varying ratio a, remains very nearly the same for the same numer µ and the same numer N, and this other property is no less worthy to oserve as well as it illustrates our entire argument excellently. Certainly I demonstrate it thus. Let y hypothesis a =, the maximum proaility = q, what the condition is when it is assumed m = N; next, for the same numer N, let there e assumed a very small inequality etween a and and under that other hypothesis let the maximum proaility e designated = Q; ut this 9 happens at the index a+ : let the difference e supposed etween the two indices N and a+, which will e = a a+ N. Thus the proaility of the term will e (under the hypothesis a = ), of which the index is indicated y = q : c a+ ) 2N, a+ 10

11 ecause namely there should e assumed for µ certainly this last proaility, if it is multiplied y ) m ( a a+ ) N; 2N 2 a+ it will give, y strength of the second paragraph, the proaility Q for the same index a+, which will have een given y the maximum proaility under the alternate hypothesis. Thus therefore there will e had Q = q c a+ ) 2 N ) ( ) m 2N 2 a + where it is supposed m = N + µ or for this work m = a+, and y this fact y sustitution there happens q ) ( ) :(a+) 2N 2 Q = c a+ ). 2 N a + Now I shall demonstrate, ut if a and differ slightly etween themselves, the factor is ale to e reckoned ) ( ) :(a+) 2N 2 = c a+ ) 2N. a + thus as Q = q is ale to e assumed; to this end there should e put = 1 and a = 1+α again y understanding y α a very small fraction; thus there happens a + = 2 + 2α (1 2 + α N = + 12 α 14 αα + 18 ) α3 N; therefore, with the hyperolic logarithms assumed, the aforementioned equality to e demonstrated vanishes in this other ( α 14 αα + 18 ) α3 N log a + 2N log 2 ( ) 2 a a + = N. a + Because if now further the values for a and are sustituted 1 + ( α and ) 1 2 and the quantities log a 2 a ; log a+ and a+ are converted into series, with the terms neglected in which α transcends the third dimension, it will produce log a, or log(1 + α) = α 1 2 αα α3 ; next finally log 2 a + = 1 2 α αα 1 24 α3 ; ( ) 2 a = 1 a + 4 αα 1 4 α3 ; 11

12 ut with these terms sustituted, if the multiplications are actually performed with the terms neglected again, in which α transcends the third dimension, the identical equation is otained perfectly. Therefore without any hesitation Q = q is ale to e reckoned. Thus our argument, which at first glance appeared greatly dark, shines forth with unexpected light; for the entire work had een put into it, that for whatever value a the index µ is counted from the term given y the maximum proaility, or that for µ the a+ : excess of oys is taken aove the numer y this fact the variations of the proailities in each example will e most nearly the same for the same numers µ assumed either affirmatively or negatively. But also proailities themselves, where they are maximums, in each example, without any scruple, the same are ale to e reckoned, if only the difference etween a and is not taken exceedingly large. Thus likewise the ratio of that quantity is apparent, of which we have made mention 10 at the end. 16. From the previous it is well-known, to the extent that the proailities are ale to e determined as nearly as possile for whatever numer of irths, for whatever numer of oys and for whatever ratio etween a and. Behold the entire process! The maximum proaility may e sought first for the hypothesis a =, which y the power of the seventh paragraph of the earlier paper = N + 1 and this proaility will remain nearly the same, when it is the maximum, for each other ratio etween a and ; indeed the maximum proaility will e ale to e put more simply = N a+, Next the numer is assumed which indicates the numer of oys delighting with maximum proaility, y which fact whatsoever other numer of oys expressed y the formula is given a+ + µ; I say the proaility to e nearly = N 1 c µµ:n. Nor do I think this thing to e ale to e made more suitale and likewise more accurate, when the numer N is great. If that numer will have een very small, the entire work with total rigor will e completed y the power of the formula expressed at the end of second paragraph. If finally the middle ones, one has seen y analysis, one wishes to prefer either formula to the other. 17. I shall illustrate the entire process y a single example selected from the tales of Süssmilch, although somewhat less suitale on account of the enormity of our numer µ: of course the celerated Süssmilch in the second part of his work, to which at the end many tales have een added, page 13 tale IV reports the example for the capital city of Austria in the year 1728, with respect to which 3102 little girls and

13 oys had een orn; therefore here N = = 2561 and put again = there happens a+ = 2651; whence µ = = 631; therefore for the most special case itself the proaility is = c 155 ; certainly that small fraction is less, than unity applied to unity prefixed to sixty-nine zeros, of which sort the smallness eludes every conception indeed if we add all proailities, in which the numer of oys is assumed to descend elow 2021, hardly thence the aforementioned proaility will e tripled; therefore if the question will e how much is the proaility that among 5122 irths the numer of oys is reduced elow 2021, I say the numer is greater than three prefixed with sixty-eight zeros, to e ale to e contested against one not to e that this happens; can it e that it will have happened in Vienna and can it e that with so many events repeated another similar monstrosity will have happened, just as the cited tale reports, others may judge. Nor is it concealed to e ale to happen, that in Vienna little daughters and also little oys are created more easily and more frequently; for in the same tale it is reported for the year 1724, the aptized girls to have een only 1422 ut 3005 little oys, any such enormous inequality contrary to the other, if again it e sujected to calculation, it will e held as morally impossile y everyone. Certainly the irregularities of such size are ale to e ascried neither to a natural law nor to chance in any way. 18. But if the entire numer of irths is very small, then the cases which appear most extraordinary, are much less improales than it is ale to e presumed; I shall set forth an example, which is certain to me. Certainly in the year 1763 in a small town of the Basil Region, of which the name is Liechstahl, 2 20 little sons and 37 girls had een orn. The distriution of irths of this kind, where the numer of girls is nearly the doule of the little oys, truly it is not ale to not e greatly rare, ut on account of the paucity of irths it holds nothing, ecause with the aforesaid examples it is ale to e compared in any manner. Behold the numerical calculation. We have namely 2N = 57 and exactly (further with a = remaining) a+ = 29.26; therefore µ = = 9.26 µµ and N = 3.01; thus c µµ:n = therefore the sought proaility = = 1 190, 2 Translator s note: This would e the present day Liestal which lies aout 20 km southeast of Basel. a 13

14 which at least it is worth for the case itself, has een such. Therefore for 57 irths it is proale that within 190 years that case most itself, such as has happened, may happen once. Because certainly it has een remarkale from the unique paucity of oys, rightly the proailities of single cases should e added, where the numer of oys is suppressed as yet more and then the sum of all these proailities ascends just aout to 1 70 ; and indeed if the numer of all annual oservations will have een 70, it has een proale that it itself may happen once ecause it has happened, certainly that from 57 irths the male offspring may e depressed elow 21; I have made the other reduction of the proaility to 1 70 in passing; furthermore I confess the numerical calculations, on account of the smallness of the numer N and the relative magnitude of the numer µ, indeed not exactly to delight with accuracy; nevertheless I contend the errors to e very slight, which are ale to e neglected easily. 19. Now we understand further, ecause y computing the numer µ from numer, at which the maximum proaility falls, the proaility is the same for the same numer µ, whatever ratio may intercede etween a and, if only this ratio recede very little from equality, we understand, it is said, that also the sum of the proailities for the same numer µ of the terms must e the same moreover in the first schedule, I have given a tale y aid of this I have determined the limits etween which, that the numer of oys may susist, the contest is equal; now I say the same limits to e ale to e assumed for whatever slightly unequal ratio etween a and if only it happens that the maximum proaility may fall in the middle of these limits. But we have seen in the last paragraph of the twelfth paragraph under the hypothesis a = and also 2N = 20000, the fact that these limits are and , which are equidistant from the numer 10000, each evidently y the numer ; therefore now I say the fact that with the ratio a to changed and with it put = , with the same numer of irths remaining, a similar condition will happen within the limits and or within the limits and equidistant from the term given y the maximum proaility ( 10) where the common distance again is Therefore in return the proaility is equal that the numer of oys may cross or not cross these limits. I have een amazed at the very narrow width of these terms. But also further in the preceding paper I have taught in the thirteenth paragraph, the distances of the limits to e diminished most nearly in suduplicate ratio of the numers, which indicate the sum of all irths. Thus for 5000 irths an equal contest will e most nearly, to e that the numer of oys is no larger than or 2591 nor less than or

15 which limits are indicated simply 2567 ± 24. If in France the total annual offspring is assumed the mean male offspring will e = and it will e just aout an equal wager, the excess or defect of the enumerated masculine offspring will not e greater than 260, if, it is compared with the middle position, with a = put evidently. 20. I hoped these inquiries concerning the equally proale limits will e useful, that a more prudent and more accurate judgment may e ale to e produced concerning the true natural law or concerning the true proportion etween the numers a and : can it e that of lands everywhere? can it e that it is constant for each time itself? can it e that all variations, which are remaining, are to e ascried to chance? can it e that natural law itself permits some variation? Even yet concerning this I hesitate: an exceedingly small difference etween a and is seen and an involvement of the influence of chance too much than that y the maximum numer of oservations is ale to e determined accurately: finally the numers themselves, which should have een ale to e oserved exactly, are not more reliale from every suspicion. The ratio a :, which I ascrie to natural law, at any rate is not ale to e deduced otherwise than out of a great numer of oservations; ut many deductions of this sort are ale to e deduced; the most simple method, y which it is assumed a to to e, as the sum of the oys orn to the sum of the young daughters orn, to me until now seems to e preferred to all the rest. Nevertheless I think the criteria not to e despised, which have een placed within limits enriched with equal proaility; here concerning the thing I shall say somewhat more eloquently. 21. Where the sum of the irths is greater, y it it is safer for determining the ratio a :. Through the entire 95 years at London little sons and girls have een orn; whence it is estalished est a = = (in the writings of Süssmilch with a small error is put see page 21.) From this mean value the oservations recede at any time notaly, although an entire ten year period is received: the ten year period produces girls and young sons and thus a = 1.040, which value is least among all periods of ten years; the maximum is for the seven year period , in which girls and males have een orn; whence a = 1.081; out of the summation of the aforesaid period of ten years and of seven years emerges a = 1.054, which value agrees enough with the ordinary hypothesis. The ten year period indicates greater aerration; for a = is put; ut an error has een committed and should e put in the place of of this method the errors y themselves reveal themselves; ut likewise of the other errors incurred, which in no way are ale to e divined, they create dread. Nevertheless let us put certainly the value a = and there should have een for the period of ten years , 2N = (the numer should have een put in place of the numer indicating the sum of male offspring ); thus it produces the numer a + = 93451, 15

16 which rejoices with maximum proaility for the numer to e indicated of male offspring; ut it has happened y lot that that numer was only 92814; therefore the aerration due to chance here has een = 637 for the entire generation ; now let us put the greater generation yet y far, just as 4,000,000; ut we have seen everywhere the aerrations for the same degree of proaility to e in suduplicate ratio of the generations; therefore the aerration now will e equally proale = = 2986: ut now there is further a + = , which numer indicates the most proale value of male offspring, with a = put and then the numer of all little daughters arises; let us sutract the error 2986 due to chance from the numer of oys and let us add the same to the numer of young daughters, we will have the numers and , for each sex, which are ale to happen with the same facility for the generation 4,000,000 as well as the numers and have happened for the generation ; ut there is : = 1000 : Therefore, if even it is certain with a = put, it will e nevertheless ale to happen in order that even in the generation of 4,000,000 infants a = may e revealed; further the error arising y chance is ale to happen in excess with the same facility and then the ratio a = may happen. Therefore the oservations of two hundred years estalished in London, although they may have een most accurate, will not yet e sufficient for removing hesitation in regard to estalishing the natural law or ratio a. Wanderings to and fro are ale to e much greater in y far fewer generations, as the tales confirm. 22. I shall add one thing concerning the middle or equally proale limits of aerrations due to chance; namely again let the numer e a + ± µ, where a + expresses the numer of oys according to the law of nature and µ the aerration; we have seen for 2N = there to e µ = and for whatever other value µ = N = N; therefore it will e equally proale that µ is greater or lesser than N and for many years the fortuitous events must answer to this law not adly, thus as one may happen nearly as often as another, not even the very small inequalities may destroy this quality of the value a. I have consulted therefore in the tale of London the period of ten years and with the calculation put I have had success almost eyond expectation. Behold the London tale defended with my calculation, where the fifth column supposes a =

17 or ut the seventh column 3 supposes and precisely a + = N; a = a + = N: Year Girls Boys Sum 2N N Aerration µ N Aerration µ NB NB NB NB NB NB NB NB NB NB NB This very small tale confirms excellently our entire theory, so clear as approached near. In the sixth column the aerration 47 has een noted y the sign NB, although such small quantity crosses the defined limits; in the sixth column the positive signs and in the eighth column the negative signs prevail, while yet the aerration to each part is ale to rise with equal facility; it itself of the extraordinary, which has happened y chance, I attriute to the efficiency of chance. Certainly I shall not delay a long time with these content with the method exposed, y which likewise several other related arguments will e ale to e treated with success. 3 Translator s note: Over the years displayed in the tale, the total numer of oys is and the total numer of girls is Hence the ratio is