Ostalo
Rogerius Joseph Boscovich / Ruđer Josip Bošković, De calculo probabilitatum quę respondent diversis valoribus summę errorum post plures observationes, quarum singulę possint esse erroneę certa quadam quantitate, The editio princeps of Bošković’s autograph in the Bancroft Library within the University of California at Berkeley
Ivica Martinović
; Dubrovnik, Hrvatska
Sažetak
Here published is the editio princeps of Bošković’s autograph De calculo probabilitatum quę respondent diversis valoribus summę errorum post plures observationes, quarum singulę possint esse erroneę certa quadam quantitate, housed in 1962 at the Bancroft Library within the University of California at Berkeley, in the collection Boscovich Papers, call number: Carton 1, Part 1: no. 62, Folder 1:79.
The transcription is accompanied by notes and introduction. The latter contains the following: (1) description of the manuscript; (2) outline of the contents of Bošković’s writing related to the theory of probability, notably the mathematical problem as formulated by Bošković; (3) description of the status of investigation, with regard to the fact that until today Oscar Sheynin, Russian science historian, has been the only scholar to study this manuscript, about which he published three articles in the 1970s. Following in these initial research steps is the investigation primarily aimed at: (4) exact dating of this undated manuscript; (5) establishment of whether the manuscript was completed or not; and, lastly, (6) assessment of its significance within Bošković’s work and within the history of science in the eighteenth century.
The mathematical problem, introduced in the title of Bošković’s manuscript as the “calculus of probabilities which correspond to various values of the sum of errors after several observations,” under the condition that single observations can differ from the exact measurement “for a given quantity,” Bošković formulated as follows: “If in a certain series of observations equally probable errors 1, 0, −1 are presupposed for single observations, we ask the ratio of probability for single sums, which sums can thence be obtained after a given number n of observations.
Error sums can take on all the values from n up to −n for different combinations. The probabilities for any value will relate as the numbers of combinations, from which the same value is obtained.
In order to determine this number for single sums, may the series of these errors be arranged in three lines:
I 1, 1, 1, 1, 1, 1, 1, 1 etc.
II 0, 0, 0, 0, 0, 0, 0, 0 etc.
III −1, −1, −1, −1, −1, −1, −1, −1 etc.
Then we ask the numbers of combinations for single values of the sum of errors from 0, 1, 2 etc. up to n. The numbers [of the combinations] for [the sums of errors] −1, −2, −3 etc. can be easily found; these numbers will be the same as the numbers for [the sum of errors] 1, 2, 3 etc., because once those [with the positive sum of errors] are found, these [with the negative sum of errors] will be found too.”1
In his preparation of the solution, Bošković introduced the notion of combination of n objects taken r at a time. With great scrutiny he then examined the first case, when the sum of errors is equal 0, and provided the formula for the number of all combinations under the condition that the sum of errors equals 0. By using the same method, he provided the formulas for the number of combinations when the sum of errors equals 1, 2, …, 8. With regard to these formulas he then examined the “determination of the number of useful terms” (determinatio numeri terminorum utilium), i.e. omitted the terms equal to 0. In the final step he applied his formulas to calculate the number of combinations for different numbers of observations n = 1, 2, …, 8. Bošković’s autograph De calculo probabilitatum is not dated, and as early as 1973 Oscar Sheynin emphasised that the significance of this manuscript rested on its dating:
“The manuscript under consideration being undated its evaluation is hardly possible. If it was written in 1750–1753, i.e. during his participation in the arc measurement, or at least before 1756 [read: before the publication of Simpson’s letter ‘On the Advantage of taking the Mean of a Number of Observations, in practical Astronomy’ in Philosophical Transactions], Boscovich should be regarded as the first to use a quantitative stochastic method in the theory of errors and, thus a precursor of T. Simpson.”2
Indeed, the text of the manuscript is not linked with the geodetic and cartographic expedition along the Roman meridian from Rome to Rimini in the period 1750–1752, or with Bošković’s statistical method for correcting discordant observations, for the first time applied in Bošković’s article “De litteraria expeditione per Pontificiam ditionem” (“On the scientific expedition through the Papal State,” 1757) in the journal of the Bologna Academy, and for the first time published in Bošković’s supplement to Book Five of Benedikt Stay’s epic Philosophia recentior on Newton’s and Bošković’s natural philosophy under the title »De recentissimis graduum dimensionibus, et figura, ac magnitudine Terrae inde derivanda« (“On the most recent measurements of longitude of meridian degree and on the shape and magnitude of the Earth derived thereof,” 1760). In 1760, while working on the sample of five geodetic observations, Bošković formulated a different problem with three conditions:
“Given the number of degrees [i.e. the number of measurements of meridian degrees], find the corrections which ought to be assigned to the individual observations, whereby the following three conditions ought to be preserved: namely, [1] that diferrences between longitudes of degree be proportional to the differences between the values of sinus versus of duplicate latitude; [2] that the sum of positive corrections be equal to the sum of the negative ones; [3] that the sum of all the corrections, either positive or negative, be the smallest of all sums which can be obtained if the first two conditions are preserved.”3
The formulation of the problem from 1760 departs considerably from that in the manuscript De calculo probabilitatum. The only thing in common is that the number n of measurements is finite, in fact, very small: in the manuscript De calculo probabilitatum n reaches 8 as the biggest value, whilst in the geodetic reports n = 5 in the first application in 1757, and n = 9 in the second one in 1770.
In order to establish the year or at least the time-frame in which Bošković wrote his De calculo probabilitatum, in this article two approaches have been employed, both based on my own research experience. The first approach is guided by the question: Did Bošković encounter the theory of errors in geophysics and geodesy, and if so, when and how, and if he did not encounter the theory of errors, was he at least familiar with the existence of different measurements of the same quantity? Bošković himself provided one answer to that question in 1760:
»365. Since 1738, in the treatise on the shape of the Earth I have expressed suspicion about the irregularity in measuring the series of degrees and the shape of the Earth, the cause of which I specifically perceive. This suspicion I pursued in the mentioned treatise, as well as in the other two treatises from 1741 and 1742.«4
In his statement, Bošković draws attention to three of his treatises: Dissertatio de Telluris figura (1739), pp. XVIII-XXIII, De inaequalitate gravitatis in diversis Terrae locis (1741), n. 37, and De observationibus astronomicis, et quo pertingat earundem certitudo (1742), n. 63. In them only has Bošković referred to the same problem: uncertainty in measuring the length of a meridian degree, and suspicion about the irregularity of the Earth.
The second approach is facilitated by the features of Bošković’s manuscript De calculo probabilitatum, which depart from the later style of Bošković’s scientific works. In August 1740, Bošković published two short treatises which document an easily discernible change in his approach to the composition of a scientific text. The treatise De motu corporum projectorum in spatio non resistente was the last printed dissertation prepared for the public defense at the Collegium Romanum that he structured as a series of propositions, in addition to which he emphasised the introductory lemmas and scholia. In the treatise De circulis osculatoribus Bošković for the first time numbered the paragraphs, i.e. divided his text into ‘numbers.’ He rarely departed from the numeration of paragraphs, for example, in the reports submitted to rulers or in the articles for some journals. In the manuscript De calculo probabilitatum Bošković resorted to a compromise, i.e. he introduced the proposition and numerated the steps to the solution by assigning centred subtitles to them.
The genesis of Bošković’s research in geophysics and geodesy on the one, and textual analysis, which suggests that the manuscript originated before the key change in the composition of Bošković’s early treatises in August 1740, on the other hand, pave the path for the following conclusion: the time-frame in which the manuscript De calculo probabilitatum was written should be narrowed to the early period 1738–1740.
This implies that, besides the already indicated two approaches―statistical method for correcting discordant observations with the first aplication in geodesy (1757, 1760, 1770), and the application of probablity in the selected topics in his natural philosophy (1754, 1755, 1758, 1763)―this manuscript provides an insight into Bošković’s earliest approach to probability, best represented by the formulation of the problem in the theory of errors at the beginning of the manuscript.
Ključne riječi
Ruđer Bošković; theory of errors; theory of probability; combinatorics; discordant observations; 18th century mathematics; geodesy; astronomy; geophysics
Hrčak ID:
222795
URI
Datum izdavanja:
17.5.2019.
Posjeta: 889 *