18th centuryIntroductionHuygens tract remained the only text on probability for 50 years. The early years of the 18th century witnessed a series of publications on probability by Montmort, Nicolaus Bernoulli, DeMoivre and posthumously Jacob Bernoulli. This might have been stimulated by 'whispers' and writings about that elusive piece Ars Conjectandi, on which Jacob Bernoulli had been brooding for 20 years, and which was still not finished when he died.After Montmort died, it was DeMoivre who reigned suppreme with his Doctrine of Chance. From the middle of the 18th century the combination of observations became an important topic that was studied by Boscovich, Laplace and others. 

1705  Jacob Bernoulli dies. A eulogy by Fontenelle which contains a summary of his Ars Conjectandi is published the following year. Due to family disputes, it will take another 8 years before the Ars Conjectandi is published. The sad part is that the main text was already finished in 1690. 
1708  Pierre Remond de Montmort publishes his Essai d'Analyse sur les Jeux de Hazards. 
1709  Nicolaus Bernoulli's dissertation De Usu Artis Conjectandi in Jure dated june 1709, is published. It contains large parts of text that are directly copied from Jacob Bernoulli's Meditationes and the Ars Conjectandi. 
1710  John Arbuthnot reads his paper An Argument for Divine Providence, taken from the constant Regularity observed in the Births of both Sexes (published 1712) to the Royal Society. He presents the number of yearly christenings for males and females for the period 16291710. He notes that there are more males then females and that the proportion is almost constant. The original part is that he then calculates the probability, given no difference in number, of this outcome which is 0.5^{82}. Extrapolating this result to ...Ages and Ages...and...all over the World he concludes ... that it is Art, not Chance, that governs. 
1711  Abraham de
Moivre
publishes his De Mensura Sortis, seu, de Probilitate Eventuum in Ludis a Casu Fortuito Pendentibus
in which he introduces the concept of statistical independence, but expresses this in ratio's of products of numbers of wins and losses.
Si eventus duo nullo modo ex se invicem pendeant, ita ut p sit numerus casuum quibus eventus primus contingere possit, & q numerus casuum quibus possit non contingere; & sit r numerus casuum quibus eventus secundus contingere possit, & s numerus casuum quibus possit noncontingere: Multiplicetur p+q per r+s & Productum Multiplicationis, viz. pr + qr + ps + qs erit numerus casuum ommnium quibus contingentia & noncontingentia eventuum inter se variari possunt.. 
1712  Willem Jacob 's Gravesande publishes his Démonstration Mathématique du soin que Dieu prend de diriger ce qui se passe dans ce monde, tiree du nombre des Garçons et des Filles qui naissent journellement. He meets with Nicolaus Bernoulli (16871759), who is visiting the Hague on his way to England, and discusses Arbuthnots paper with him. 's Gravesande improves upon Arbuthnots approach by correcting for the differences in number of births that occur each year. 
1713  Jacob Bernoulli's
Ars Conjectandi, posthumously published by his nephew Nicolaus Bernoulli,
contained four parts

1713/14  Pierre Remond de Montmort publishes a second, enlarged edition of Essai d'Analyse sur les Jeux de Hazards. 
1714  William Browne publishes his translation of Huygens' De Ratiociniis in Ludo Alae. He abandons his initial plan to add an extra part with examples, because as he writes in the preface, this is now sufficiently covered by Pierre Remond de Montmort's enlarged Essai ... A second edition of John Arbuthnot's translation of 1692 is published the same year. 
1718  Abraham de
Moivre defines statistical independence (again) in his first edition of Doctrine
of Chances, using probabilities
... if a Fraction expresses the Probability of an Event, and another Fraction the Probability of an another Event, and those two Events are independent; the Probability that both those Events will Happen, will be the Product of those Fractions. (tDoC, p. 4) 
1730  Abraham de Moivre publishes the central limit theorem in the special case of the binomial distribution 
1733  Abraham de
Moivre shows in his Approximatio ad Summam
Terminorum Binomii (a + b)^{n} in Seriem expansi. the normal distribution to be an approximation of the
binomial distribution. His conclusion:
And thus in all cases it will be found, that altho' Chance produces irregularities, still the Odds will be infinitely great, that in process of Time, those Irregularities will bear no proportion to the recurrency of that Order which naturally results from original Design. (English translation from tDoC 1738, p.243)We have to wait 77 years before the normal distribution is recognized by Gauss and Laplace as the universal description of how observational errors are distributed 
1738  Abraham de Moivre publishes the second, enlarged edition of Doctrine of Chances, with an expanded translation of the Approximatio (tDoC 1738, pp. 235243) 
According to Stigler (1986) around 1750 the advantage of combining observations
slowly becomes clear. Until then the notion exists that when combining
observations, errors will multiply instead of compensate. An exception
is the 16th century Danish astronomer Tycho
Brahe as described by Hald (1990).
In 1749 Leonhard Euler while trying to solve the problem of the inequality in the motion of Jupiter and Saturn, is not inclined to combine observations. Tobias Mayer while tackling a similar problem takes this conceptual barrier and solves the problem 

1757  The Dalmatian jesuit Roger Boscovich publishes his ideas on combining observations in a synopsis of the 1755 publication with his English confrère Christopher Maire on the measurements of a meridian arc near Rome. A full description of his method, to be published in 1760, will later be included in another publication with Maire Voyage astronomique et geographique dans l'état de l'église (1770). 
1763  Posthumously Thomas Bayes theorem is presented, but is hardly noticed on the continent until 1780 
1774  Pierre Simon Laplace
publishes his Mémoire sur la probabilité des causes par les évènemens
in which he tries to Determine the mean one should take among three given
observations of the same phenomenon. It is based on an unpublished memoir from 1772 and the publication seems largely motivated by the knowledge that others (JosephLouis Lagrange and Johan III Bernoulli) are working on the same problem. A wrong turn in his approach however leaves Laplace stranded with an equation of the fifteenth degree as the solution to this problem (Stigler 1986; p. 105, 116). 
1787  Pierre Simon Laplace publishes his Théorie de Jupiter et Saturne in which he solves the problem of the inequality in the motion of Jupiter and Saturn and proves the stability of the solar system. He improves upon the method used by Tobias Mayer to combine observations. 