Opyt geomtrii o chetyrekh izmereniakh. Geometriia sinteticheskaia ("An Essay on the Geometry of Four Dimensions. Synthetic Geometry")

A FORGOTTEN WORK ON FOUR-DIMENSIONAL GEOMETRY BY AN UKRAINIAN HERO

8vo (250x165 mm). [6], 150, [4] pp. and one large folding plate with geometrical figures. Editor's printed wrappers (worn, marginal paper loss, spine broken and roughly repaired). Some quires loose. A genuine copy preserved in a cloth box.

Incredibly rare first and only edition of this pioneristic work on four-dimensional geometry.

Mykola Ivanovich Hulak (or Nikolai Ivanovich Gulak or Goulak) was born in Warsaw in 1821 into a noble, but poor family from the Zolotonish district of the Poltava province. He spent his childhood and youth in the village of Mykolaivka, in the Kherson region. After graduating from the local gymnasium, he studied law at the University of Tartu in Estonia, graduating in 1844. From 1845 he served the Kiev governor-general as a translator and archaeographer.

Hulak played an active part in the founding of the Cyril and Methodius Brotherhood (cf. P.S. Goncharuk, The harbinger of friendship and cooperation of peoples: to the 180^th anniversary of the birth of Cyril and Methodius M.I. Hulak, Kyiv, 2002). He belonged to the radical wing of the organization, which defended national-democratic ideals and the idea of a peasant revolution. He advocated the elimination of autocracy and serfdom, the abolition of estates, universal education, and the unification of Slavic people. In January 1847 he moved to St. Petersburg, where he was arrested in March for his political activism. During interrogation, he refused to testify or name any of the brotherhood's members. He spent three years in the Shlisselburg Fortress, after which he was exiled to Perm. It was only 12 years later that he returned to Ukraine and began teaching. He taught geography and mathematics at the Russian-language lyceum in Odessa (1859-1861), history at the Kushnikov Women's Institute in Kerch (1861-1862), mathematics at the Stavropol gymnasium (1862-1863), physics and cosmography at the Kutaisi Gymnasium (1863-1867), and mathematics, physics, Latin and Greek at the Tiflis Gymnasium (1867-1886). After retiring, he moved to the Azerbaijani city of Yelizavetpol (later Kirovabad, today Ganja), where he died. A street in that city is named after him.

Hulak had been interested in mathematics even before the Brotherhood was founded. While in prison, he studied the subject and later carried out research primarily related to non-Euclidean geometries and four-dimensional space. In 1859, he published “Studies on Transcendental Equations” in Odessa, and in 1877, he published the present work in Tbilisi (cf. M. Briefly & O. Antonyuk, Mikola Gulak as a mathematician: Monograph, Lutsk, 2004).

In particolar, Hulak's second book aroused great interest. He published it with a dedication “To the memory of Lobachevsky”. He considered Lobachevsky's discoveries in non-Euclidean geometry to be of revolutionary importance. The introduction to the book is titled “A Conversation about Space”. Written in the form of a dialogue between the author and his friend “V”, it contains an exchange of thoughts about the theories of Nicolaus Copernicus and Nikolai Lobachevsky. The book, one of the first ever published in this field, deals with synthetic geometry and explores the properties of figures in a four-dimensional space. Hulak invented neologisms for the new concepts he was explaining (such as ‘sphero-sphere', ‘plano-planes', ‘prismo-prisms', ‘pyro-pyramides', so forth), and was one of the first to systematically treat four-dimensional geometry in a synthetic way, anticipating by decades some of the best-known developments in the West (even though his works remained little known outside certain circles). Hulak also wrote a continuation to the “Attempt at Four-Dimensional Geometry” entitled “Analytical Geometry with Four Dimensions and with Spherical Tetrahedronometry”, but this work remained unpublished. The original manuscript ended up in the collection of Professor I.K. Andronov (cf. I.K. Omelchenko, Gulak, Mykola Ivanovich, in: “Great Ukrainian Encyclopedia”, online; G. Bevz, Mykola Hulak, in: “Mathematics at School”, 2011, no. 9, pp. 46-47).

Hulak (pp. 25-26) claims that he had reached his conclusions in 1876, completely independently of Carl Friedrich Zöllner's Prinzipien einer elektrodynamischer Theorie der Materie (‘Principles of an Electrodynamic Theory of Matter', 1876): ‘Such a coincidence of my thoughts with those of a renowned scientist can only bring me joy because it confirms that we are both addressing a genuinely advanced issue that aligns with modern science'.

During the Soviet era, Hulak was completely forgotten. It is only in recent years that he and his work have been rediscovered. He himself was quite an impressive figure. He was the first Ukrainian orientalist. He spoke Ukrainian, Russian, German, French, Latin, Greek and several Oriental languages (cf. Z.K. Aliyeva, Intelligentsia in linguistic studies and Georgian-Azerbaijani-Ukrainian relations, in: “Scientific notes of the Nizhyn State University named after Mykola Gogol. Series: Philological Sciences”, 2013, 4, pp. 302-308). Roman Ivanychuk's 1984 historical novel, “The Fourth Dimension”, is dedicated to Hulak and his passion for mathematics. In the afterword to the novel, Ivanychuk writes: “Mykola Ivanovich Hulak was the most active organiser of the Cyril and Methodius Brotherhood. He was an outstanding mathematician, lawyer, literary scholar, orientalist, translator and philosopher with encyclopaedic knowledge. He was also a political scientist who knew twenty languages perfectly”.

“Interestingly, [the renowned Russian polymath and religious philosopher Pavel] Florensky understood it [the concept of the fourth dimension] not only as time, but also as ‘depth' -a peculiar quasi-metaphysical depth of reality that should not be confused with depth in a usual, three-dimensional sense, and especially not with perspectival depth. This rather uncommon notion came from a book called An Essay on the Geometry of Four Dimensions. Synthetic Geometry (“Opyt geometrii o chetyrekh izmereniyah. Geometriya sinteticheskaya”, 1877) by a Ukrainian scholar and educator named Nikolai Goulak (alternatively spelled as Mykola Hulak), whose biography Florensky considered writing during his university years. In the introduction, echoing Johann Carl Friedrich Zöllner's reflections on the non-Euclidian character of cosmic space, Goulak suggested that space in the universe may be most accurately described through Bernhard Riemann's elliptic geometry and conceived of as a closed ‘sphero-sphere' (sharo-shar) with constant positive curvature. ‘And having once assumed that our space has constant positive curvature, in all its points, we thus allow the existence of the fourth dimension-depth', which is ‘imperceptible by our sensory organs, but nevertheless as real as length, width, and height' (Goulak 1877, 13-14). The following exchange between the author (‘I') and his fictional interlocutor (‘V') is particularly instructive: ‘V. All this is wonderful, but I can't understand what exactly this fourth dimension-depthis, and why isn't it perceptible by our sensory organs? After all, you said yourself that the direction of depth is as real as the directions of length, width, and height […] I. Indeed, and I even claim that the depth axis is mutually perpendicular to the other axes and makes 90-degree angles with them. V. Well, here's the point O in space: I will pass through it, in fact or in imagination, three mutually perpendicular axes: length, width, and height, and you pass through this point, mentally, the fourth axis of depth. I. With pleasure. First of all, notice that the depth axis has, like the other three axes, two opposite directions: a positive direction and a negative direction. Consequently, this axis will pass through the point O, and at this point -but only at this one point- will cross our space […] Departing from the point O in the positive direction of the depth axis, we will leave our space and deepen into the sphero-sphere, for which Riemannian space serves as the outer edge. Take a good look at the ex-pression: deepen. So, the direction of the fourth axis lies not to the right or left, not above or below, not at the front or behind the point O, but straight inside it. Following the positive direction of the depth axis, we will finally reach the center of the universe […] (Goulak 1877, 20-21). Further on, to account for the inability of the human senses to perceive this mysterious ‘axis of depth', Goulak employed a popular explanatory device -the analogy with imaginary sentient inhabitants of two-dimensional space who would have no idea of a higher dimension. ‘Similarly, in our material world of three dimensions, we see and perceive not the very essence of external objects, but only their three-dimensional projection, just as in a picture we see their projection in two dimensions' (Goulak 1877, 21). In support of this argument, he cited several acclaimed sources, such as Plato's Republic, Goethe's Faust, and Louis Claude de Saint-Martin's Of Errors and Truth (Des erreurs et de la vérité, 1775). But the most important text (referenced even on the cover of the book), where the word ‘depth' seems to have originated, was a verse from Apostle Paul's epistle to the Ephesians (3:18). In his article on ‘The Meaning of Idealism' (Smysl idealisma, 1915), Florensky acknowledged Goulak as the first person to notice this verse (in the geometrical context) and cited it along with several previous ones: ‘I bow my knees before the Father of our Lord Jesus Christ... May He grant you, -writes St. Apostle to the Ephesians,-... to be strengthened with His Spirit in the inner man, and may Christ dwell in your hearts through faith, so that you, being rooted and grounded in love, could comprehend with all the saints what is the breadth and length and depth and height'. Florensky could recall this passage (which directly connects the concept of the ‘inner man' to four dimensions), and the obscure book on ‘synthetic geometry' along with it, thanks to the work of Pyotr Ouspensky, the main Russian theorist of the fourth dimension and promoter of Charles Howard Hinton's ideas. In the conclusion to his Tertium Organum (1911), discussing the words of St. Paul, ‘startling by reason of their mathematical exactness', Ouspensky remarked that they not merely connect sanctity to knowledge, but imply ‘that sanctity gives a new understanding of space'. Florensky borrowed this idea and, in a way, turned it against Ouspensky: criticizing ‘the methods of distillation of new capabilities', devised by Hinton and popularized by his Russian counterpart, as ‘unnatural' and ‘violent' to the ‘spiritual organism', he described the fourth dimension as ‘the depth of the world, which can be perceived only by a righteous soul'. This four-dimensional depth, therefore, served as a kind of alternative to the habitual three-dimensional one, which Hermann von Helmholtz had shown to be a psychological phenomenon produced by the so-called ‘unconscious inference'. For Florensky, the latter seems to have represented a by-product of the fallen human mind that lost the capacity to see the world in its original and supertemporal unity. Identifying Goulak's axis of depth with the axis of time, he explained that, being seen properly, ‘any reality is stretched in the direction of time no less than it is stretched in each of the three directions of space', which means that it has a certain ‘time depth' and ‘constitutes some kind of formation of the four-dimensional geometry, i.e. not a body, but a super-body, or, in N.[I.] Goulak's terminology, body-body [telo-tela]': ‘Now, any reality extends in four dimensions and is isolated in an image of four dimensions. And time, the fourth coordinate of this image, is organized in it as its own [internal] time […] This time is not an external time, which is the time of lifeless things devoid of strong individuality. And that is why the time of a given image cannot be measured by the time of other images […]: it is necessary either to enter into the proper time of a given image and consider it as a self-contained unity or to ascend by contemplation to an image [of a higher order] that concretely encompasses this image and others […] Then this new image, in relation to those particular images, will act as their common space, with its own special time, i.e. the four-dimensional space, and they, these particular images, in relation to the common image, will be the objects within it, connected between themselves by power and energy interaction' ” (P. Yushin, The Cover of the Journal Makovets and Pavel Florensky's World Picture, in: “In anderen Zeiten/In Other Times. Zeitdiskurse im Wandel/Changing Discourses of Time across Human History”, B. Hohenwald, N. Illner & J. Renn, eds., Berlin, 2022, pp. 170-174).

We were not able to trace a copy of this book in any public library worldwide.

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