Historical texts · Translations · Gauss editions

Carl Friedrich Gauss — original-language collected works

The current Theoria motus reader is Gauss_VII_Original_Language_Native002_178a.pdf, containing articles 1–130 in 173 pages. The PDFs and editable source packages listed below are attached to this collection.

This collection contains original-language readers and editable LaTeX for Gauss’s writings in arithmetic, algebra, analysis, geometry, probability, geodesy, mechanics, magnetism and astronomy. The full transcription and its subsequent English translation remain unfinished. The scope of each reader is given below.

Volume V covers physical leaves 2–122, 201–291, 351–542, the Schumacher letter beginning within leaf 546 through leaf 574 before the Erläuterungen, and 586–648. Its historical contents also name writings outside this selection. The Allgemeine Theorie des Erdmagnetismus and attached maps, the intervening unavailable work, the Copenhagen/Mailand graphical report, and the later Erläuterungen with charts are outside this reader.

Volume VI follows the supplied collected edition. It does not supply original periodical passages, ephemerides, maps, catalogues or the Iris calculation absent from that printing. Volume VII contains the complete first book and the opening of the second book through article 130 in the upper part of native leaf 178. Its 173 pages comprise 172 pages of historical content and one modern authorship and source notice. It includes aberration, light-travel time, inverse coordinate conversion and differential relations for geocentric position, with the complete numerical example and its formulation through the orbital elements. The third section presents trigonometric relations, the determination of a conic orbit from two radii and a given semiparameter, perihelion direction or eccentricity, and the determination from three radii. It then treats the problem of determining an orbit from two radii and the elapsed time, including an iterative method and the complete approximation derivation, formulas I–VIII and the historical Euler citations, followed by two complete numerical examples comparing the precision of those approximations, and the elliptical relations between two orbital places, with formulas 1–12 and four starred counterparts, followed by an auxiliary-angle transformation and the formulas for l and L, the series and continued-fraction expansions of the auxiliary expression, the coefficient laws, both historical notes and the range of the third auxiliary table. The cubic equations, successive approximation and choice of the applicable root follow, with the historical notes and the construction and interpolation of the second auxiliary table. The trial solution for larger auxiliary angles and the possible multiplicity when complete revolutions intervene are followed by the determination of orbital elements and anomalies from the auxiliary angle, with formulas for the semiaxes, parameter, eccentricity and epoch. Further half-angle formulas give the orbital elements with their ambiguity conditions and calculation checks. Three complete numerical examples determine the orbital elements and anomalies, followed by the perihelion-time calculation and its precision qualifications. The parabolic case includes its complete derivation and the determination of the elements. The hyperbolic case includes the relations between two orbital places, the auxiliary quantities, their series and continued-fraction expansions, the equations used to determine the hyperbolic arc, the recovery of orbital elements and perihelion time, a complete numerical example for a hyperbolic orbit, and the elapsed-time formula in terms of the semimajor axis, sum of radii and chord, with its historical attribution and ambiguity discussion. The following articles present the elapsed-time series and the independent parabolic derivation, with both historical manuscript notes on chord geometry, time and perihelion distance. The third section concludes with the hyperbolic elapsed-time derivation and both series. The fourth section opens with the determination of the orbital plane from two heliocentric places using longitude and latitude or rectangular coordinates, including ambiguity conditions, projected areas and the choice of reference plane. Article 112 relates three heliocentric positions through projected triangle areas and gives the complete equations in geocentric and heliocentric quantities. Article 113 introduces the abbreviated three-place expressions, their signed permutations and complete list, and their interpretation through the volume of a pyramid. Article 114 gives the elimination formulas for three geocentric distances and their exceptional cases, concluding the fourth section and first book. The second book opens with the data required to determine an orbit from geocentric observations and the exceptional ecliptic-plane case. Articles 116–117 give the reduction of observations to a fixed reference plane and three treatments of parallax. Articles 118–122 give the treatments of aberration, the reduction to two implicit equations, and the approximation and iteration procedures for determining their unknowns, with the historical note on the order of the omitted terms. Articles 123–128 treat the choice and spacing of observations and five methods using two assumed distances. They compare geocentric positions, heliocentric positions, elapsed times, pairs of orbital elements or quantities calculated before the elements themselves, retaining their nodal, small-arc and near-parabolic limitations and precision considerations. The complete fifth method includes its logarithmic comparison and final choice of observations for determining the elements. Article 129 gives five corresponding methods using inclination and ascending-node longitude, with their nodal restrictions. Article 130 sets out the three requirements for an initial orbit-determination method and introduces the preliminary considerations. Its source is the 1906 collected printing of the 1809 Theoria motus.

Historical words, spellings, formulas, numerical values, notes and references remain identifiable in their source sequence. Mathematical irregularities and historical errata are not silently reconciled. The text, formulas, tables and included mathematical drawings are editable LaTeX; each historical portrait is an image. The source packages contain the build inputs for the corresponding reader. Modern fonts, reflow and vector drawing conventions do not reproduce historical type or ink contours. Damaged and ambiguous readings retain qualifications, and no error-free, human-reviewed or independently reviewed edition is claimed.

In Volume VII, article 77 uses a recurring seven-shaped sign for daily mean motion. The editable italic seven-shaped glyph represents its observed contour, not the numerical value seven; its alphabetic identity remains unresolved. The source package records this reading qualification. Modern fonts do not reproduce historical type identity.

In the first equation II of article 83, a small detached point between the plus sign and the following half-factor is represented at its observed raised position. Its presence is visible, while punctuation identity and printing intent remain uncertain; incidental ink, damaged type and printing or imaging irregularity remain possible. The faint alternative witness does not resolve that intent. No additional mathematical meaning is assigned to the point. The source package retains the associated reading qualifications.

In article 86, the exponent after 243a in the corrected pi numerator is represented as 4 with moderate glyph confidence. Dense ink obscures its interior; nearby numeral comparisons, the retained earlier reading and the mathematical context support 4, while the faint alternative scan does not resolve every stroke. The alternatives 2 or damaged or merged type remain documented in the source package. Four terminal commas and three terminal points retain their observed fraction-bar height without additional mathematical meanings.

In article 87, the second leader mark following 2 log cos 2 omega is represented as a comma from its compact upper body and distinct downward curved tail. Its shape is clear, while printing intent remains uncertain; incidental ink, damaged type, printing or imaging irregularity remain possible. No additional mathematical meaning is assigned to it. The source package retains this reading qualification.

In article 88, the closing parenthesis in formula 10 is represented from separated upper and lower curved strokes. Its identity is recognizable, while the cause of the gap is uncertain. Two terminal points and the comma following the inline square-root expression for a retain their raised positions without additional mathematical meaning. Damaged type and printing or imaging irregularity remain possible. The source package retains these reading qualifications.

In article 89, the comma separating the final formulas for l and L and the terminal point retain their positions at fraction-bar height. Their contours and positions are clear, while printing intent remains uncertain. No additional mathematical meaning is assigned to these marks; incidental ink, damaged type and printing or imaging irregularity remain possible. The fourth root, both square roots, all signs and the original exponent positions are retained.

In article 90, three commas and three terminal points retain their positions at fraction-bar height, including two points in the historical note. Their contours and positions are clear, while printing intent remains uncertain. No additional mathematical meaning is assigned to these marks. A thin detached diagonal below the first g-cubed is not confidently a linguistic or mathematical sign. Incidental ink, damaged type and printing or imaging irregularity remain possible. The four continued fractions retain every printed coefficient and sign, including the final 40/255 in the second expression for xi; both historical notes are included.

In articles 91–92, four terminal points, three formula commas and one formula semicolon retain their positions at fraction-bar height. Their contours and positions are clear, while punctuation identity and printing intent remain uncertain; no additional mathematical meaning is assigned. A detached multicomponent mark before “prima” in article 91 is not confidently linguistic or mathematical. The closing parenthesis after H in the cubic equation of article 92, the opening parenthesis in its radical expression, and the closing parenthesis after “tabulam II” in article 93 have interrupted contours. Editable parentheses represent the recognizable glyphs without asserting continuous ink. Incidental ink, damaged type and printing or imaging irregularity remain possible. The historical notes are retained as printed, including the four root cases and the final phi inequality.

In article 94, the comma following the trigonometric fraction retains its position at fraction-bar height. Its contour and position are clear, while punctuation identity and printing intent remain uncertain; no additional mathematical meaning is assigned. Incidental ink, damaged type and printing or imaging irregularity remain possible. The denominator retains the printed exponent 3 after g.

In article 95, six terminal points and six formula commas retain their positions near fraction-bar height. A detached stroke above omega in the phrase introducto angulo is represented by a grave, and a detached point between nota and erit retains its raised position. These contours and positions are clear; their punctuation or accent identities and printing intent remain uncertain. No additional mathematical meaning is assigned. Incidental ink, damaged type and printing or imaging irregularity remain possible. Both X-weighted area expressions retain the printed exponent 3/2.

In article 96, the terminal points of formulas 27 and 30 and four commas following fractions retain their raised positions. Their contours and positions are clear; punctuation identity and printing intent remain qualified. Small irregular marks above the second cosine in formula 28 and beside two fourth-root signs, and a short stroke below the sine baseline in formula III, are not confidently identified as separate mathematical signs. Incidental ink, damaged type and printing or imaging irregularity remain possible. No additional mathematical meaning is assigned.

In article 97, short horizontal strokes above the g in the log sin E row and above the e in diverso are represented by editable macrons. Their contours and positions are clear; their identity as intentional macrons and their printing intent remain qualified. No additional mathematical or linguistic meaning is assigned. Other small irregular marks near numerical values, an equality sign and a logarithm label are not confidently identified as separate signs. Incidental ink, damaged type and printing or imaging irregularity remain possible. The numerical values, omitted units and differences between printed calculations are retained without recalculation.

In article 98, the exponent after sin f in equation 3 is represented as 3. The principal scan has a dense, 8-like contour; the corresponding alternative Internet Archive scan shows a recognizable 3, and the preceding formula also supports that reading. The principal glyph remains uncertain, with 8, damaged or filled type, and the recovered transcription’s 2 retained as alternatives. This is a qualified reading, not a claim of a clearly visible 3 or a historical printing error. Small detached points after suppositionibus and between the second x equals sign and zero are represented at their observed heights. Two terminal points and a comma are also raised to their observed fraction-bar positions. Their presence and positions are clear; punctuation identity and printing intent remain qualified. No additional linguistic or mathematical meaning is assigned. Incidental ink, damaged type and printing or imaging irregularity remain possible. The alternative scan is faint and does not establish every symbol independently or the physical absence of the small points.

Alternative witness for article 98: Internet Archive, Werke volume 7, item werkehrsgvonderg07gausuoft, Gerstein–University of Toronto digitization; retained PDF werkehrsgvonderg07gausuoft.pdf, leaves 140–141, printed pages 130–131. This witness supports the qualified exponent reading and the comparison of the small detached marks.

Alternative witness for article 124: Internet Archive, Werke volume 7, item werkehrsgvonderg07gausuoft, Gerstein–University of Toronto digitization; PDF werkehrsgvonderg07gausuoft.pdf, leaf 175, printed page 165. This witness supplies the comparison for the qualified unprimed X reading beside the small detached contour in the principal witness.

In article 99, six terminal points are represented at fraction-bar height, and a detached point after the first quotient in the eliminated time equation is represented below the baseline. In article 100, the comma after the defining fraction for Z is raised, and a detached point above and to the right of the 8 in the third-power numerator is represented by an offset raised point. Their presence and positions are clear; punctuation identity and printing intent remain qualified. No additional mathematical or linguistic meaning is assigned. Incidental ink, damaged type and printing or imaging irregularity remain possible. Modern fonts and reflow represent the content rather than historical type identity.

In article 102, the closing parenthesis after habeat is represented from separated upper and lower curved strokes. The recognizable glyph is clear; the cause of the interrupted contour remains uncertain. Small detached marks near the equations and text in articles 101–103 are not confidently identified as linguistic or mathematical signs. Incidental ink, damaged type and printing or imaging irregularity remain possible. Modern fonts and reflow represent the content rather than historical type identity.

In article 104, four terminal points are represented at the height of the adjacent fraction bars, without assigning a multiplication meaning. Their presence and position are clear; their punctuation identity and printing intent remain less certain. Small detached marks near itaque and confirmationem are not confidently identified as linguistic signs. Incidental ink, damaged type and printing or imaging irregularity remain possible. Modern fonts and reflow represent content rather than historical type identity.

In article 105, the first leader after log tang 2n is represented by a comma-shaped mark followed by eleven individual points. Its contour and position are clear; its printing intent remains uncertain. Small detached marks near log(1+z), the value 8,9848318, the right margin and the logarithm row are not confidently identified as linguistic or mathematical signs. In article 106, the terminal point of the four-term arc-cosine formula is represented at fraction-bar height without assigning a multiplication meaning. Its position is clear; punctuation identity and printing intent remain less certain. Detached marks near spectantibus and reliquas are not confidently identified as linguistic signs. Incidental ink, damaged type and printing or imaging irregularity remain possible. Modern fonts and reflow represent content rather than historical type identity.

In article 107, small irregular marks below several series product points are not confidently identified as separate signs. The final terminal point is represented at fraction-bar height without assigning a multiplication meaning. In article 108, the first historical manuscript note retains the printed coefficient one half, and both notes retain their unmarked form. Three terminal points are represented at fraction-bar height; the first note ends with an ordinary period. Their positions are clear; punctuation identity and printing intent remain less certain. Small detached marks below two formulas are not confidently identified as linguistic or mathematical signs. Incidental ink, damaged type and printing or imaging irregularity remain possible. Modern fonts and reflow represent content rather than historical type identity.

In article 109, Greek α and Latin a remain distinct, including the stated substitution of −α for a. Three terminal points are represented at fraction-bar height without assigning multiplication meanings. A small detached stroke near “sequentem” is not confidently a linguistic accent. In article 110, two terminal points are represented at fraction-bar height, and a detached short horizontal stroke below one ascending-node symbol is represented by an editable stroke. Its presence and position are clear; its identity and printing intent remain uncertain, and no mathematical meaning is assigned. Small marks near “possit” and “per formulas”, and a small detached contour between two coordinate equations in article 111, are not confidently separate linguistic or mathematical signs. Incidental ink, damaged type and printing or imaging irregularity remain possible. Modern fonts and reflow represent content rather than historical type identity.

In article 112, a tiny detached irregular horizontal mark above the gap after the first D in the first cosine equation is not confidently identified as a separate linguistic or mathematical sign. Its presence and position are clear; printing intent remains uncertain. Incidental ink, damaged type and printing or imaging irregularity remain possible. No additional mathematical meaning is assigned. Modern fonts and reflow represent content rather than historical type identity.

In article 113, Arabic 0, 1 and 2 are distinct from the upright Roman O, I and II in the abbreviated expressions. The zero relation in the great-circle argument has a two-bar equals sign. A small detached angular mark above its right end, a low irregular mark beside the first beta in the opening formula, and detached blobs above words in the concluding paragraph are not confidently identified as additional linguistic or mathematical signs. Their contours and positions are visible; printing intent remains uncertain. Incidental ink, damaged type and printing or imaging irregularity remain possible.

In article 114, the first multiplier contains the tangent of capital B double-prime. Arabic 0, 1 and 2 remain distinct from upright Roman O, I and II. A detached short angular mark below the first row of equation 8 and a short rising mark below the word hic in the concluding paragraph are not confidently identified as additional mathematical or linguistic signs. Their contours and positions are visible; printing intent remains uncertain. Incidental ink, damaged type and printing or imaging irregularity remain possible.

In articles 115–117, detached marks occur between quod and in, beside the i-dot of ubi, above aberrationem and fixam, and above and below the end of hanc. Their contours and positions are visible, but they are not confidently identified as additional linguistic signs. Printing intent remains uncertain; incidental ink, damaged type and printing or imaging irregularity remain possible. Kepleri retains its small-capital base and upright lowercase final i. The phrase cum cunctis quam optime conspirent in article 115, and obliquitatis mediae, commodius and obliquitatis apparentis in article 116, are upright.

In article 118, a detached angular contour below verus in the third listed relation is not confidently identified as an additional sign. In article 119, small detached contours beside the Y-equation and below aequationum are likewise uncertain. Article 120 retains a terminal point and a comma at fraction-bar height; article 122 retains three commas at that height after its parenthesized differential quotients. These positions convey no additional mathematical meaning. Irregular contours attached to the first opening parenthesis in article 122 are not identified as separate characters. The contours and positions are visible, while punctuation identity or printing intent remains qualified; incidental ink, damaged type and printing or imaging irregularity remain possible. Greek alpha and Latin a, and the xi, eta, lunate epsilon, zeta and theta forms, remain distinct. The historical three-paragraph structure of article 120 and its statement quatuor systematibus are retained.

In article 124, a short detached upright contour lies above and to the right of X in the phrase differentiarum X,Y. Its presence and position are clear, but its character identity and printing intent are uncertain. It is shorter than the nearby definite prime used for comparison. The ordinary unprimed reading X,Y is retained with support from the alternative witness and the defined quantities; the contextual argument is an inference, not observed ink. A damaged prime, incidental ink, damaged type, and printing or imaging irregularity remain alternatives. No physical absence of ink or independence of typesetting or image processing is asserted. Articles 123 and 124 retain their respective one and four historical paragraphs, all observation limits, and the complete exceptional case involving two successive intervals near 180 degrees.

In article 126, a tiny two-component contour appears above and to the right of the final m in arcum. Its presence and position are clear, but it is not confidently identified as independent linguistic punctuation or an accent. Incidental ink, damaged type, and printing or imaging irregularity remain possible. In article 127, the point following the reciprocal 1/a is represented at its observed fraction-bar height without assigning multiplication meaning. Its position is clear; punctuation identity and printing intent remain qualified. Articles 125–127 retain their respective one, two and one historical paragraphs, the full limits on their methods and the complete concluding comparison with the fifth method. Modern editable fonts represent the content and positions, not historical type identity; no physical absence of ink, human review or independent review is asserted.

In article 128, the terminal points after the second square-root relation, the three-member equality and the Y expression, and the comma after the p fraction, are represented at their observed fraction-bar height without adding mathematical meaning. Their positions are clear; punctuation identity and printing intent remain qualified. Small detached irregular contours above the third initial expression, below ut, to the left of the p fraction and beside primae are not confidently identified as linguistic or mathematical signs. Incidental ink, damaged type and printing or imaging irregularity remain possible. The full fifth method retains both historical paragraphs and all final alternatives. The long equality reflows without changing any member, factor or order. Modern fonts represent content rather than historical type identity.

Article 129 preserves the distinct printed spellings tantummodo and tamtummodo, the five method paragraphs and the complete concluding nodal restrictions. Article 130 retains both historical paragraphs and the italic words Primo, Secundo and tertio. Small detached irregular contours above the left part of q in quod, above-left of c in censeri and below n in desiderandum are not confidently identified as linguistic accents or mathematical signs. Their presence and positions are clear; printing intent remains uncertain. Incidental ink, damaged type and printing or imaging irregularity remain possible. Modern fonts and reflow represent content rather than historical type identity.

Historical author: Carl Friedrich Gauss. The historical editors, correspondents, contributors and issuing institutions retain their printed authorship and attributions, including the editorial material signed by Schering and Fricke. The sources are Werke I (1863), II (1876, second impression), III (1866), IV (1873), V (1867), VI (1874), VII (1906) and VIII (1900). Volume I also uses the separately identified 1870 second printing for comparative readings. Volume VIII uses a complete alternative Kyushu photograph for the right-hand text of historical page 95 cut off in the principal scan.

AI-transcribed text and AI-produced corrections and typesetting incorporate OpenAI GPT-6.1 Sol (Ultra) and GPT-6 Astra (Ultra) work in Volume I; the current source comparisons and the modern transcription and drawing work in the other supplied volumes use OpenAI GPT-6 Astra (Ultra). Earlier ChatGPT packet drafts do not have an established exact backend identity. These credits concern modern encoding and do not replace the authorship of inherited writing. The per-volume RIGHTS_AND_CREDITS.md files accompany the editable sources.

Historical provenance: Volume I’s GDZ witness, PPN235993352, Volume VII’s GDZ witness, PPN236008730, and the earlier deposited collection. The historical Gauss deposit family identifies that deposit history, not this GitHub collection. No DOI is asserted for these GitHub files.

These editions are freely available for study and teaching.