Nikolai Lobachevsky – Migoro Media

Nikolai Lobachevsky: The Mathematician Who Bent the Rules of Geometry

A pioneer of non-Euclidean space who redefined the shape of mathematical thought

Nikolai Ivanovich Lobachevsky (1792–1856) was a revolutionary figure in the history of mathematics — a quiet disruptor who dared to question a 2,000-year-old assumption at the heart of geometry. Known today as “the founder of non-Euclidean geometry,” Lobachevsky’s work reshaped our understanding of space and laid essential groundwork for modern mathematics and theoretical physics, including Einstein’s general theory of relativity. In an age when Euclidean geometry was regarded as the only possible description of physical space, Lobachevsky posed a radical question: What if the parallel postulate — the idea that only one line can be drawn parallel to a given line through a point — was not necessarily true?

His audacious response created a new kind of geometry, one where space curves and parallel lines multiply.

“There is no branch of mathematics, however abstract, which may not someday be applied to phenomena of the real world.”
— Lobachevsky (attributed)

Despite facing skepticism and near-total obscurity during his lifetime, Lobachevsky’s contributions would eventually earn him a place among the giants of mathematical thought. His journey from provincial Russia to intellectual legacy is a powerful story of perseverance, originality, and scientific courage.

Keywords: Lobachevsky biography, Nikolai Lobachevsky, non-Euclidean geometry, Russian mathematician, history of mathematics, hyperbolic geometry

🧒 Early Life and Education

Nikolai Ivanovich Lobachevsky was born on December 1, 1792 (November 20, Old Style) in Nizhny Novgorod, a city on the Volga River in the Russian Empire. He came from a middle-class Russian Orthodox family. His father, Ivan Maksimovich Lobachevsky, was a minor government official, and his mother, Praskovia Alexandrovna, was a homemaker. After the death of his father in 1800, the family moved to Kazan, a city that would become central to Lobachevsky’s academic life and career.

Education Begins in Kazan
In 1802, at just ten years old, Lobachevsky was enrolled in the Kazan Gymnasium, where he began to show promise in mathematics and science. In 1807, he entered the newly established Imperial Kazan University — one of the first universities in Russia east of Moscow — where he studied physics, astronomy, philosophy, and classical languages.

A Key Influence: Johann Bartels
Among his professors was Johann Christian Martin Bartels, a German mathematician and former close associate of Carl Friedrich Gauss. Bartels introduced Lobachevsky to the rigorous style of mathematical thinking that would shape his future. Bartels emphasized the importance of questioning assumptions and using logic to uncover truth — a mindset that would empower Lobachevsky to challenge the very foundations of Euclidean geometry.

During his student years, Lobachevsky became known for his intellectual independence and his willingness to explore controversial or abstract ideas. He graduated with a degree in physics and mathematics in 1811, at the age of 19, and was appointed a lecturer at Kazan University just three years later.

This early period laid the groundwork for one of the most revolutionary mathematical careers in history — not just through formal education, but through exposure to progressive European scientific thought and mentorship rooted in critical inquiry.

🎓 Academic Career and Rise in Kazan

After graduating from Kazan University in 1811, Nikolai Lobachevsky quickly distinguished himself as more than just a brilliant student. By 1814, at the age of 21, he was appointed a lecturer in mathematics at the same university — a rare and prestigious achievement at such a young age.

Climbing the Academic Ladder
Lobachevsky’s academic progression was swift. He became an extraordinary professor in 1816 and a full professor by 1822, owing to his rigorous teaching methods and growing reputation for intellectual daring. He taught mathematics, physics, and astronomy, often challenging students to think critically and reject blind adherence to classical doctrine.

Rector of Kazan University (1827–1846)
In 1827, Lobachevsky was appointed Rector of Kazan University, a position he held for nearly two decades. As rector, he implemented wide-ranging reforms to modernize the university. He restructured the curriculum to reflect advancements in mathematics and the natural sciences, established laboratories, expanded the university library, and recruited qualified faculty from across Europe and Russia.

His goal was clear: to transform Kazan University into a center of scientific excellence in the Russian Empire. Under his leadership, the university became a beacon for mathematical research and education — especially in a time when Russian higher education was still heavily influenced by outdated scholastic traditions.

Reformer vs. Traditionalists
Despite his efforts, Lobachevsky faced considerable resistance from conservative academics and administrators who viewed his reforms as dangerous or disruptive. His nonconformist teaching style, controversial mathematical views, and management decisions made him a polarizing figure. Internal university politics became increasingly strained during his later years as rector, ultimately contributing to his forced retirement in 1846.

Yet even amid opposition, Lobachevsky remained committed to his vision of a university grounded in scientific inquiry, innovation, and critical reasoning. His tenure at Kazan not only shaped the institution’s future but also provided him the academic freedom to pursue what would become his most important contribution to mathematics: non-Euclidean geometry.

📐 The Birth of Non-Euclidean Geometry

For over two millennia, Euclid’s Elements stood as the unshakable foundation of geometry. Yet at its core lay a persistent puzzle: the fifth postulate, also known as the parallel postulate. Unlike Euclid’s other axioms, the fifth was awkward and less intuitive, stating (roughly) that only one line parallel to a given line can be drawn through a point not on that line. For centuries, mathematicians tried to derive it from the other axioms — and failed.

🧠 A Revolutionary Idea
Nikolai Lobachevsky took a radically different approach. Instead of attempting to prove the parallel postulate, he asked:
What if it isn’t true? What kind of geometry would emerge without it?

This single act of mathematical daring led Lobachevsky to independently construct a new geometry, later known as hyperbolic geometry, in which:

An infinite number of lines can be drawn parallel to a given line through a point not on it.
The sum of angles in a triangle is less than 180 degrees.
No similarity exists between different-sized triangles unless they are congruent.

📄 “Imaginary Geometry” Published
Lobachevsky first introduced his ideas under the name “imaginary geometry” in a series of articles published in the Kazan Messenger (Казанский Вестник) between 1829 and 1830. These writings were groundbreaking, but they went largely unnoticed — partly due to the regional obscurity of Kazan and partly because the content was so radical that many simply dismissed it.

In 1840, Lobachevsky attempted to reach a wider European audience by publishing a more formal exposition in French:
📘 Geometrische Untersuchungen zur Theorie der Parallellinien
(Geometrical Studies on the Theory of Parallels)
This publication laid out his theory in more rigorous detail and with clearer language for continental scholars — yet again, the reception was muted.

🧾 Key Concepts Introduced:

Hyperbolic geometry: A consistent geometry where Euclid’s parallel postulate is replaced with its opposite.
Acute angle hypothesis: The assumption that the sum of angles in a triangle is always less than 180°, leading to a new kind of space.
Multiple parallels: Through any point outside a line, there exist infinitely many lines that do not intersect the original — i.e., they are all parallel in hyperbolic space.

🚫 Initial Rejection and Isolation
Lobachevsky’s ideas were ahead of their time. The mathematical community, steeped in Euclidean tradition and philosophical realism, largely ignored or rejected his theory. Many regarded “imaginary geometry” as a meaningless abstraction, failing to appreciate its logical consistency and revolutionary implications.

Even his own peers at Kazan viewed his work with skepticism. Lobachevsky was essentially working in isolation — with no awareness that Carl Friedrich Gauss had developed similar ideas privately, though never published them out of fear of controversy.

Despite the lack of immediate recognition, Lobachevsky never wavered in his belief that mathematics must be driven by internal logic rather than philosophical assumption. His pioneering work not only proved non-Euclidean geometry was logically consistent, but also that alternative geometries were possible — a profound shift in the foundations of mathematics.

🔁 Intersections with Gauss and Beltrami

Gauss’s Silent Support
Although Nikolai Lobachevsky faced near-total indifference from his contemporaries, one of the greatest minds of the age had quietly arrived at similar conclusions: Carl Friedrich Gauss — often called the Prince of Mathematicians. By the early 19th century, Gauss had independently explored the consequences of denying Euclid’s fifth postulate and found a consistent alternative geometry, remarkably similar to what Lobachevsky would later publish.

However, Gauss never made his discoveries public. Fearing backlash from the conservative academic establishment and concerned about the perception of “anti-Euclidean” ideas, he chose to remain silent. In a private letter from 1846, Gauss referred to Lobachevsky’s work as that of a “genius of the first rank”, though the two never corresponded directly.

Lobachevsky’s Isolation
Lobachevsky had no knowledge of Gauss’s parallel investigations and worked in virtual isolation. The silence from the European academic community only reinforced his obscurity. While Gauss admired Lobachevsky, he never publicly supported him, and this lack of endorsement meant the Russian mathematician’s groundbreaking work went mostly unrecognized in his lifetime.

Beltrami and Mathematical Vindication
Decades later, Lobachevsky’s ideas were finally validated — notably by Eugenio Beltrami, an Italian mathematician. In 1868, Beltrami constructed a concrete model of Lobachevskian (hyperbolic) geometry using surfaces of constant negative curvature. This was the first time non-Euclidean geometry was shown to be logically consistent with Euclidean principles, using an internal model.

Recognition by Klein and Poincaré
Soon after, Felix Klein and Henri Poincaré further extended the legitimacy and importance of non-Euclidean geometry:

Klein incorporated Lobachevsky’s ideas into his Erlangen Program, unifying various geometries under the lens of symmetry and transformation groups.
Poincaré, one of the greatest mathematicians of the late 19th century, applied hyperbolic geometry to complex analysis and number theory. He even remarked that non-Euclidean geometry was essential for understanding the universe, foreshadowing its role in physics.

From Theory to the Cosmos: Influence on Einstein
Lobachevsky’s legacy reached far beyond pure mathematics. His radical ideas about the structure of space became crucial to the development of Riemannian geometry, a generalization of non-Euclidean principles. This geometric framework later became the mathematical foundation for Albert Einstein’s general theory of relativity in the early 20th century.

Einstein’s model of curved space-time relies on precisely the kind of alternative geometric thinking that Lobachevsky had dared to formalize nearly a century earlier.

From Ignored to Immortal
Though largely unacknowledged in his own lifetime, Lobachevsky’s influence has since become profound and far-reaching. He laid the conceptual groundwork for a revolution in both mathematics and physics — a quiet spark that, over time, ignited the reshaping of modern science.

🧮 Other Scientific Contributions

While Nikolai Lobachevsky is best known for founding non-Euclidean geometry, his intellectual range extended across multiple branches of mathematics and science. He was not just a theorist, but a committed educator, administrator, and applied scientist who made meaningful contributions to both academic institutions and the Russian scientific community.

Contributions to Algebra and Analysis
Beyond geometry, Lobachevsky conducted research in algebra and mathematical analysis. He explored topics such as roots of algebraic equations, iterative methods for solving equations, and series convergence. His work demonstrated a focus on practical problem-solving methods — an approach that mirrored his interest in teaching real-world applications of abstract concepts.

He also developed a numerical approximation method (now known as the Lobachevsky method) for finding real roots of polynomial equations. Though not widely adopted today, it was an early attempt at numerical root-finding techniques still relevant to computational mathematics.

Astronomical Work
As a professor and later head of the astronomy department at Kazan University, Lobachevsky contributed to celestial mechanics and observational astronomy. He organized and improved the university’s observatory, helping to advance practical scientific research in Russia. His observational work included planetary motion, comet tracking, and the refinement of instruments used in measuring stellar positions.

Educational Reformer and Administrator
As Rector of Kazan University for nearly 20 years, Lobachevsky was a tireless advocate for scientific education. He:

Promoted experimental science and mathematics over classical rhetoric
Updated curricula to reflect contemporary European science
Advocated for student-centered learning and clear teaching standards
Emphasized academic meritocracy, regardless of a student’s social background

He also supervised construction of new buildings, upgraded the library, and enhanced the university’s laboratories — laying the foundation for Kazan University’s reputation as a scientific institution.

Creation of Mathematical Terminology in Russian
Lobachevsky was deeply committed to making mathematics accessible to Russian speakers. At a time when much of scientific work in Russia was still conducted in Latin, French, or German, he coined and standardized mathematical terms in the Russian language, helping to establish a native mathematical lexicon that could be used in education, publishing, and public discourse.

Probability and Mechanics
In addition to his core research, Lobachevsky published papers on probability theory, discussing its philosophical foundations and practical relevance to everyday problems — from insurance to risk analysis. He also explored mechanics, particularly the motion of bodies under forces, integrating mathematical methods with physical intuition.

A Polymath with a Purpose
These works reflect a thinker who was not only speculative and theoretical but also applied and pragmatic. Lobachevsky believed that mathematics should serve society — a belief that guided his administrative decisions, teaching methods, and scientific priorities.

🧔 Personal Life and Character

Though Nikolai Lobachevsky is remembered primarily for his groundbreaking mathematical work, his personal life reveals a portrait of a deeply devoted, principled, and resilient individual — shaped by hardship, duty, and an unshakable belief in the power of education.

Marriage and Family
In the mid-1830s, Lobachevsky married Varvara Alexeyevna Moiseyeva, a woman from a well-regarded family in Kazan. The couple had seven children, and Lobachevsky was known to be a loving father who prioritized his family, even as he maintained his heavy teaching, research, and administrative responsibilities. Though personal details about his family life are scarce in official records, his correspondence suggests he held deep affection for his wife and children.

A Scholar of Strong Convictions
Lobachevsky was widely respected — and sometimes feared — for his sharp intellect and strong personality. Described by colleagues and students as modest and unpretentious, he was nevertheless known to be combative in academic debates. He did not shy away from challenging prevailing opinions, especially in matters of curriculum reform and scientific methodology.

He was particularly devoted to Kazan University, often referring to it as a “sacred institution.” Even while serving in administrative roles, he continued to teach, publish, and mentor students personally. His lectures were known for their clarity, energy, and refusal to adhere to rote memorization — earning him a loyal student following.

Dedication Amid Hardship
In his later years, Lobachevsky endured a series of personal and physical hardships. Most notably, he suffered from progressive blindness, which severely limited his ability to write and conduct research. Despite his deteriorating vision, he continued to work through dictation and the assistance of colleagues and family. His final mathematical writings, though less prolific than his earlier work, were completed under conditions of near-total blindness — a testament to his perseverance.

A Quiet Decline
After being removed as rector in 1846, partly due to mounting opposition from conservative university factions and partly because of his declining health, Lobachevsky remained largely isolated from the broader scientific community. His final years were spent in modest conditions, with only sporadic recognition of his contributions. Yet he remained dignified and committed to his ideals until his death.

A Legacy Beyond Genius
To those who knew him, Lobachevsky was not just a pioneering mathematician, but a man of integrity, passion, and resilience. His character — shaped by early hardship, intellectual courage, and unwavering commitment to education — left an impression that outlasted his lifetime.

⚰️ Decline and Death

A Life Dimmed by Darkness
In the final decade of his life, Nikolai Lobachevsky faced a series of difficult personal and professional setbacks. Chief among them was the gradual loss of his vision — a condition likely exacerbated by years of overwork, poor health, and the limited medical care available at the time. By the early 1850s, he was nearly blind, relying on others to read and write for him.

Retirement from Public Service
In 1846, Lobachevsky was formally dismissed from his post as Rector of Kazan University, marking the end of his two-decade-long administrative career. While officially attributed to health concerns, the decision was also influenced by academic politics and ongoing resistance from conservative factions who had long opposed his reforms and progressive views.

Working Through the Shadows
Despite his disability, Lobachevsky did not retreat entirely from academic life. He continued to teach and conduct research for as long as he was physically and mentally able. With assistance from colleagues and family members, he dictated his final manuscripts — though much of his later work remained unpublished or incomplete due to his condition and isolation.

Final Days
Nikolai Lobachevsky died on February 24, 1856 (February 12, Old Style), in Kazan, Russia, at the age of 63. He passed away in relative obscurity, with little fanfare or recognition outside his immediate academic circle. At the time of his death, his greatest achievement — the invention of non-Euclidean geometry — remained largely unacknowledged by the broader scientific community.

Legacy in Waiting
Though his death was quiet, history would not forget him. Within a few decades, Lobachevsky’s ideas would be rediscovered, validated, and celebrated, securing his place among the most visionary thinkers in the history of mathematics.

🏛️ Legacy and Recognition

Rediscovered Genius
Although Nikolai Lobachevsky died in relative obscurity, the decades that followed saw a dramatic reappraisal of his work. In the late 19th century, as mathematicians like Eugenio Beltrami, Felix Klein, and Henri Poincaré validated and expanded the ideas of non-Euclidean geometry, Lobachevsky was rightfully recognized as one of its founding pioneers. What had once been dismissed as “imaginary” became central to modern mathematical and physical theory.

Celestial Tribute
In recognition of his lasting contribution to science, the International Astronomical Union named a lunar crater “Lobachevsky”, located on the far side of the Moon. This symbolic gesture placed his name quite literally among the stars — a fitting tribute for someone whose ideas forever expanded our concept of space.

The Lobachevsky Prize
In 1896, forty years after his death, the Russian Academy of Sciences established the Lobachevsky Prize, one of the most prestigious awards in the field of geometry. It honors mathematicians whose work reflects the kind of originality, rigor, and vision that Lobachevsky exemplified. Past recipients have included some of the most influential geometers of the modern era.

Educational and Institutional Honor
Lobachevsky’s name has since been engraved in academic history:

Kazan Federal University, where he spent most of his life, proudly commemorates him with statues, plaques, and an institute bearing his name.
His theories are now taught globally in advanced mathematics courses.
He is regularly featured in textbooks, biographies, and university lectures as a case study in intellectual courage and conceptual innovation.

Founder of Modern Geometry
Lobachevsky is now universally regarded as one of the founders of modern geometry, alongside János Bolyai (who made similar discoveries independently) and Bernhard Riemann. His work challenged the exclusivity of Euclidean space, proving that alternative, logically consistent geometries could exist.

This insight laid the groundwork for:

Riemannian geometry — used in general relativity
Topological and differential geometry
Modern understanding of curved space, complex manifolds, and cosmology

Impact Beyond Mathematics
The implications of Lobachevsky’s work extended far beyond pure math. His exploration of space and form had profound effects on:

Physics — especially relativity and cosmology
Philosophy — in debates over the nature of truth, reality, and the role of logic in science
Mathematical Pluralism — showing that more than one consistent mathematical universe could exist

Immortality Through Ideas
Nikolai Lobachevsky’s story is one of vision triumphing over neglect, and of truth emerging through time. His name now stands beside those of Gauss, Euler, and Einstein — not just for what he proved, but for how he dared to reimagine the structure of space itself.

🕰️ Historical Context and Influence

A Turning Point in Mathematical Thought
Nikolai Lobachevsky emerged at a pivotal moment in mathematical history — during the early 19th century, when scholars were beginning to re-examine the very foundations of mathematics. For over two thousand years, Euclidean geometry had reigned unchallenged as both a mathematical system and a description of physical space. Euclid’s Elements were considered absolute — as much a philosophical truth as a mathematical one.

Lobachevsky’s bold challenge to the parallel postulate not only introduced a new geometric framework but marked a decisive break from the axiomatic certainty of classical mathematics. His work opened the door to mathematical pluralism, where multiple internally consistent systems could exist — each with its own set of assumptions and truths.

Contrasting Euclidean and Hyperbolic Geometry
In Euclidean geometry:

Parallel lines never meet.
The angles in a triangle always sum to 180°.
Space is flat and uniform.

In Lobachevskian (hyperbolic) geometry:

There are infinitely many parallels through a point not on a line.
The sum of triangle angles is less than 180°.
Space has negative curvature — it “spreads out” more than flat space.

This wasn’t just an abstract exercise — it meant space could be fundamentally different than it appears, depending on which axioms one accepted.

Influence on Giants of Science and Math
Lobachevsky’s work would eventually influence several of the greatest minds of the 19th and 20th centuries:

Bernhard Riemann built on Lobachevsky’s idea of curved space to develop Riemannian geometry, introducing positive curvature and more general manifolds.
David Hilbert, a key figure in formalizing mathematical logic, credited Lobachevsky for breaking the psychological barrier that prevented questioning Euclidean “truths.”
Henri Poincaré applied non-Euclidean geometry to complex analysis and physics, helping integrate it into broader mathematical and scientific practice.
Albert Einstein used the principles of non-Euclidean and Riemannian geometry as the mathematical foundation of his general theory of relativity, which describes how mass and energy warp space and time.

A Philosophical Earthquake
Lobachevsky’s geometry didn’t just change math — it shook the philosophy of knowledge. Until then, many thinkers (especially following Kant) believed Euclidean geometry was a necessary condition of human thought — an innate form of intuition. Lobachevsky’s work proved that mathematics was not dictated by physical reality, but by logical structure. This helped shift philosophy toward formalism, relativism, and later, constructivism.

Mathematics as a Model, Not a Mirror
His breakthrough introduced the idea that geometry — and mathematics more broadly — is a model, not a mirror, of reality. It doesn’t reveal one “true” universe, but instead gives us multiple consistent ways to describe possible universes. That idea is now central to fields from quantum physics to computer science, topology, and cosmology.

The Beginning of Modern Mathematics
Historians now view Lobachevsky as a key transitional figure between classical and modern mathematics. By freeing geometry from its traditional constraints, he helped launch the exploration of abstract structures, alternative logics, and new forms of mathematical reasoning — laying a foundation that modern mathematics still builds upon.

📚 References and Further Reading

This section provides a curated list of primary sources, academic references, and recommended materials for students, educators, and researchers interested in the life and legacy of Nikolai Lobachevsky. All sources have been selected for their historical accuracy, academic credibility, and accessibility.

Primary Sources by Nikolai Lobachevsky

Lobachevsky, N.I. (1840). Geometrische Untersuchungen zur Theorie der Parallellinien. Berlin.
(“Geometrical Researches on the Theory of Parallels” — original French edition with German reprint)
Available in English translation via Dover Publications and Project Gutenberg.
Lobachevsky, N.I. (1829–1830). “О началах геометрии” (On the Principles of Geometry),
Published in Kazan Messenger (Казанский Вестник).
Russian original archives available at Kazan University Library.
Lobachevsky, N.I. (1834). New Foundations of Geometry with a Complete Theory of Parallels.
(Manuscript fragments translated posthumously)

Secondary Academic Works

Gray, Jeremy J. (1989). Ideas of Space: Euclidean, Non-Euclidean, and Relativistic. Oxford University Press.
An authoritative source on the evolution of geometric thought, including Lobachevsky’s role.
Bonola, Roberto (1912). Non-Euclidean Geometry: A Critical and Historical Study of Its Development.
Dover Publications (English translation by H.S. Carslaw).
Includes an accessible overview of Lobachevsky’s and Bolyai’s geometries.
Greenberg, Marvin Jay (2007). Euclidean and Non-Euclidean Geometries: Development and History (4th ed.).
W.H. Freeman.
Widely used textbook offering historical and mathematical context.
Stillwell, John (2004). Mathematics and Its History. Springer.
Includes chapters on the development of geometry and mathematical pluralism.

Institutional and Archival Resources

Kazan Federal University Archives
The university maintains a collection of Lobachevsky’s manuscripts, administrative records, and academic correspondence.
https://kpfu.ru/eng
Russian Academy of Sciences – Lobachevsky Prize Archive
http://www.ras.ru
Includes prize history, laureates, and documentation of Lobachevsky’s contributions.

❓ Frequently Asked Questions (FAQs)

What is Lobachevsky famous for?
Nikolai Lobachevsky is most famous for founding non-Euclidean geometry, a groundbreaking mathematical system that challenged the long-held dominance of Euclidean geometry. By rejecting the traditional parallel postulate, he introduced a new vision of space — hyperbolic geometry — in which an infinite number of parallel lines can pass through a single point. His work laid the foundation for modern geometry, mathematical pluralism, and even Einstein’s theory of general relativity.

What is non-Euclidean geometry?
Non-Euclidean geometry refers to any form of geometry that does not obey Euclid’s fifth postulate, also known as the parallel postulate. In hyperbolic geometry (Lobachevsky’s version), space has negative curvature, and:

Triangles have angle sums of less than 180°
Multiple parallel lines can pass through a point external to a given line
This opened the door to the idea that space itself could be curved, with vast implications in both math and physics.

Why was Lobachevsky’s work ignored during his lifetime?
Lobachevsky’s ideas were largely ignored for several reasons:

They contradicted over 2,000 years of accepted geometric doctrine
He published mostly in Russian and obscure regional journals, limiting his international reach
The mathematical community at the time was philosophically committed to Euclidean geometry
Gauss, who had privately developed similar ideas, declined to support Lobachevsky publicly
It wasn’t until the late 19th century that his work was recognized as revolutionary and mathematically valid.

Did Lobachevsky and Gauss ever correspond?
No, Lobachevsky and Gauss never corresponded directly. Although Gauss read and admired Lobachevsky’s work — calling him a “genius of the first rank” in private letters — he never publicly supported or communicated with him. Gauss feared professional backlash for endorsing such controversial ideas and kept his own non-Euclidean insights largely private during his lifetime.

How did Lobachevsky influence Einstein?
Lobachevsky’s invention of non-Euclidean geometry laid the mathematical groundwork for later developments in geometry, including Riemannian geometry, which describes curved space. Albert Einstein used Riemannian geometry to build his general theory of relativity, which describes how mass and energy warp space-time. Without Lobachevsky’s bold rejection of Euclid’s parallel postulate, the concept of curved space-time — central to modern physics — might never have taken root.