**[Mathematics]. Kinckhuysen, G. (1625-1666).**(Collection of mathematical texts). Haarlem, 1634-?, pen and black ink, occas. w. calligraphed divisional title-pages and illustrations in pen and coloured ink, contemp. vellum, sm. folio.

Gerard Kinckhuysen (±1625-1666), was a merchant and mathematician from Haarlem, who married into the wealthy family of Elisabeth Ampe (1625-1707). Her father was the successful businessman and brewer Abraham Ampe. The Ampe family were the owners of the estate that is nowadays known as *Duin en Kruidberg* (near Velsen). In the early 1630s a survey of the estate was drawn up by the surveyor and mathematician Pieter Wils. The map that he drew of the estate is held by the Velsen archive. Interestingly enough Pieter Wils was the mathematician under whose guidance the young Kinckhuysen studied mathematics. When in 1647 Pieter Wils died, his widow asked Kinckhuysen to sort his papers. This resulted in the publication of P. Wils, *Wiskonstighe wercken* (Haarlem, 1648). As is shown by the present manuscript collection of mathematical texts, Gerard Kinckhuysen must have been a precocious child. The first entry dates from 1634, when he was 9 or 10 years old. A short preface by Kinckhuysen, worth quoting in full, gives a valuable insight into the nature of the manuscript and the circumstances under which the manuscript was written:

*"Nota. oft ijmant ander Als den schrijver dit Boeck ter handen quam die moet weeten, dat dit Boeck moet aengemerckt worden alst inderdaet is. Volgt derhalve de Verclaering: Terwijl den schrijver schoolgingh heeft hij een deel dingen aengeteekent die hij int school leerde ende noch daer beneffens heeft hij hier en daer wat geschreven uijt eenigs boeck, soo dat ten lesten sonder eenigs orders[?] dit Boeck bij één geraeckt is, niettemin sooment maer tot sijn eijgen gebruijck neemt, sal men der noch veel nutticheijt uit konnen haelen."*

The contents of the manuscript are as follows:

(__ 1__). (Without title), preceded by a leaf with calligraphed: "Gerrit Kinckhuijs Anno 1634 Den 1 november"; the manuscript dated in top margin "1634" and "1635", 1-56 lvs. -

**ON:**arithmetic and interest (a few pages w. interest tables); mainly examples, problems and/ or exercises.

(__ 2__).

*"Hier volghet eene corte ende claere Instructie vanden alder cunstrijcksten Regulen d'Algebre oft Coss Anno 1635 In Haerlem den 14 martij"*(Here follows a concise exposition of the cossic algebra (...)). 57-89 lvs. -

**ON:**root extractions (square and higher roots) and basics of the Cossic algebra of the period; mainly worked examples and/ or problems, excercises. Contains a few references to other mathematicians such as Valentin Mennher, Ludolph van Ceulen, Simon Jacobs.

(__ 3__).

*"Hier Volghet De konst der zeevaert Anno 1635 in Haerlem den 24 April"*(here follows the art of navigation). 90-99 lvs. -

**ON:**some basic knowledge of astronomy for the navigator, and dead reckoning calculations. One drawing of a compass rose and several diagrams for finding latitude. Incomplete.

(__ 4__).

*"Extractio in binomische getallen"*(extraction of binomial roots). (1635), 101-102 lvs. -

**ON:**reduction of nested roots with geometric justification of the calculations.

(__ 5__).

*"Vanden hemelschen ende den Aertschen globus"*(of heavenly and terrestrial globes ). N.d., 103-106 lvs. -

**ON:**mainly latitudes and longitudes of various places on earth. With a nice drawing of a globe in pen and ink.

(__ 6__).

*"Regula Quantitates"*. 1637, (107-115) lvs. -

**ON:**algebraic problems (many Diophantine), solved by cossic algebra.

(__ 7__).

*"Cubic Coss".*1636, (116-120) lvs. -

**ON:**some problems and solutions involving equations of the third degree; the last problem references the second part of

*Florum Algebraicorum*(see below).

(__ 8__).

*"De Vier Eerste Boecken d'Algebre Van Diophante d'Alexandrie. Anno 1639 in Haerlem"*(The first four books of algebra by Diophante of Alexandria). 1639, (121-163) lvs. -

**ON:**Diophantine problems with solutions by the cossic algebra of the time. Source unknown; maybe partly Xylander's

*Diophanti Alexandrini Rerum arithmeticarum libri sex*(1575); Xylander is named in the text.

(__ 9__). (Without title, preceded by a calligraphed leaf: "Par moy Gerrardt Kinckhuijssen Anno. 1637"). 1637, (164-236) lvs. -

**ON:**"Italiaens Boeckhouden" (double entry bookkeeping), divided in several parts: "Memoriael", "Journael", "Groot Boeck", "Oncost Boeck" and "Factuer Boeck". At first sight this represents a real bookkeeping of a real merchant, but given Kinckhuysens most probable age of about 12 years old it is more likely to have been made up for the purpose of teaching.

(__ 10__).

*"Florum Algebraicorum Partis Primae".*1639, (237-285) lvs. (the first 38 leaves are not numbered, the following leaves are numbered from 1 up to 30), ills. -

**ON:**mainly problems with solutions by the cossic algebra of the time. No work with this title is known and also no subsequent parts, but see the remark above.

(__ 11__).

*"Practica Geometrica Verdeelt In Tien Boecken. Anno 1638. Begonnen 6 januari geeyndiget in 4 maenden. In Haerlem"*(The Practice of Geometry in ten books (...)). 1638, (286-388) lvs. (numbering of the first 86 leaves continued from the previous part: from 31-117 leaves, from there on unnumb.), ills. -

**ON:**plane geometry, not necessarily of practical applicability, not structured after Euclid's

*Elements*. Some nice drawings. Some problems were later published in P. Wils,

*Wis-konstige Wercken*(1648) or are taken from L. van Ceulen,

*De Arithmetische en Geometrische Fondamenten*(1615).

(__ 12__).

*"De Hondert Questien van M.r. S. Hansen*.

*Anno 1639"*(The hunderd problems by S. Hansen). 1639, (389-421) lvs., ills. -

**ON:**abbreviated solutions of 100 geometrical problems. Incl. nice drawings. The

*Hondert geometrische questien met hare solutien*(One hundred geometrical problems with their solutions, 1614) by S.H. Cardinael (1578-1647) was a popular work at the time. The solutions in the manuscript are essentially the same as those by Cardinael. See M.H. Sitters,

*Sybrand Hansz Cardinael, Rekenmeester en Wiskundige*(Hilversum, 2008).

(__ 13__).

*"De Corpora Metria oft Lichaem Metinghe"*(On the measurement of solids). N.d., (422-427) leaves, ills. -

**ON:**rules for volumes of solids. Incl. the volumes of regular solids. Probably inspired by Book 25 (among others) of Petrus Ramus,

*Meetkonst*(Amst., 1622) (Kinckhuysen mentions Ramus' name).

(__ 14__).

*"De theorie der Fortificatie ofte Sterck Bouwinge. 1640"*(The theory of fortification). 1640, (428-441) leaves, ills. -

**ON:**notes on fortresses built on the basis of regular polygons, with 4-12 sides. The subject was also studied in the Netherlands by Adriaan Adriaansz. Metius (1570-1635), by his father Adriaan Anthonisz. (1527-1607) and by Simon Stevin (c. 1548-1620), among others.

(__ 15__).

*"Institutione Astronomicae*[sic]

*. 1640"*(Foundation of Astronomy). 1640, (442-463) lvs., ills. -

**ON:**notes about various subjects in astronomy, incl. some astrology and a horoscope for 9 december 1570. The material is mostly from Adriaan Adriaansz. Metius (1570-1635) and can also be found in Metius'

*Opera Omnia Astronomicae*(Amst., 1633), e.g. Part 1, Books 1-2; the horoscope is also taken from the work of Metius, it is for Metius' own date of birth.

(__ 16__).

*"Corte aanteijckeninge van de Bewegingh der Planeeten"*(Short note on the motion of the planets). N.d., (464-468) lvs. -

**ON:**models for the motion of the sun, moon and planets with the sizes of the models. The planetary models are geocentric (Tychoic) with a deferent and a double epicycle. With positions for January 1, 1639, midnight. The lunar model is incomplete.

(__ 17__). (Spherical triangles with applications). N.d., (469 verso-507) lvs., ills. -

**ON:**the section begins with

*Triangulorum Sphericorum Rectangulorum Praxes*(Practice of right-angled spherical triangles), n.d., 14p., followed by

*Triangulorum Sphaericorum Obliquangulorum Calcus*(Computation of non right-angles spherical triangles). Standard material on the solutions of spherical triangles of all types, followed by many examples, also taken from astronomy and navigation. Appendix on spherical polygons. Explanation of prostaphairesis. Incl. tables for the altitude of the sun for every hour for a latitude of 52 degrees. At least one section of the material was taken from Adriaan Adriaansz. Metius,

*Opera Omnia Astronimica*, Part 4 Book 1. Kinckhuysen must also have used other unidetified sources.

(__ 18__).

*"Van de Sonnewysers"*(On Sundials). N.d., (508-558, (559-564 blank), 555-556) lvs., ills. -

**ON:**constructions of different types of sundials, by geometrical and numerical means. Not completed: after 102 pages of text a new subject is announced ("Eijnde der Vaste Sonnewijsers. Ende beginnen De Hangende"), but the next 6 leaves are blank. Then follow 2 leaves of tables. Most of this material taken from Adriaan Adriaansz. Metius,

*Opera Omnia Astronomica*, Part 1 Book 4 and Part 4 Book 2.

(__ 19__).

*"Vande Schaduwen Door Kegel Sneden, Ende Andere Werckingen"*(On the shadows by conic sections and other workings). N.d., (557-575, (576 blank), 577-580, (581-583 blank)) leaves. Leaves 577-580 headed:

*"volgen Eenige Vertoogen De Schaduwe Der Son Aengaende"*(here follow several discourses concerning the shadow of the sun). N.d., (4 leaves). -

**ON:**concise and elementary theory of sections of a cone and cylinder, with proofs, and application to the shadows of the sun in its daily course. Not completed. The approach is the same as in Apollonius of Perga, Conic Sections.

(__ 20__).

*"De letter Algebra Ofte Beginselen Der Wisconst"*(Symbolic algebra and the foundations of mathematics). N.d., (584-612) leaves. -

**ON:**elementary treatise on algebra in Descartes' notation. Kinckhuysen published at a later time a much improved and extended version as

*Algebra ofte Stelkunst beschreven tot dienst van de Leerlinghen*(Haarlem, 1661). On the importance of this work for Newton (who, at the instigation of John Collins, had read and annotated the Latin translation by Nicolaus Mercator of this work by Kinckhuysen), Van der Aa writes the following: "Dit werk werd zoo zeer geacht bij Newton, dat hij het niet alleen vlijtig bestudeerde, maar ook vele vraagstukken met de woorden des schrijvers en geheel op diens wijze, in zijn

*Arithmetica Universalis*zijn opgelost; ja, zoozeer was hij met Kinckhuyzen ingenomen, dat hij diens

*Algebra*in het Engelsch heeft vertaald, welke vertaling zeker het licht zou hebben gezien, ware het handschrift, met vele van Newton's papieren, niet eene prooi der vlammen geworden".

(__ 21__).

*Francisci Vieta Zeteticorum*(Of the works of Zetetics by Francois Viète). N.d., (613-635) lvs. -

**ON:**Dutch version of Francois Viète, Five books of zetetics, in Cartesian notation. Originally published Tours 1591 of 1593. Latin version (in Viète's own notation) published on p. 42-81 of Frans van Schooten (ed.),

*Francisci Vietae Opera Mathematica*(Leyden, 1646). No other Dutch version of the work is known. Cf. for the English translation R. Witmer,

*Francois Viète, The Analytic Art*(New York, 2006, p.83-158)).

- Dampstained in inner margin and partly in upper margin; occas. partlty sl. short in top margin (affecting letters, mostly of the date). Vellum dustsoiled.

= Cf. Hoogendoorn p.547ff listing 6 works published by Kinckhuysen. SEE ILLUSTRATION PLATE LXXIV.