It is regarded as a masterpiece when it comes to the application of logic in mathematics. Elements has also influenced various scientific fields, including those related to the exact sciences. Among them were Galileo Galilei, Sir Isaac Newton, Johannes Kepler, and Nicolaus Copernicus. Some philosophers and mathematicians have tried to create their own versions of Elements by incorporating the concepts introduced in the book. These include Bertrand Russell, Baruch Spinoza, and Alfred North Whitehead. They used the deductive structures that were initially created by Euclid. When the quadrivium was added to the university's curriculum, students were required to have at least some knowledge of Elements to enroll. Until the 20th century, when the content of this book was covered in other textbooks, it was regarded as something that everyone had read at least once in their lifetime. 

The constructive methods and the principles introduced by Euclid in Elements were significant. Numerous propositions demonstrate the existence of an object by providing a clear explanation of the steps involved in constructing it. In particular, his first and third postulates prove the existence of a circle and line. His previous works stated that circles and lines can be formed using prior definitions. However, in his later works, he changed his position and claimed that it was possible to construct a circle or line using an earlier proposition. He also used these figures in his proofs. For instance, in his proof of the Pythagorean theorem, he first drew a square on the side of a right triangle and then constructing it on a given line. 

The foundations of mathematics were solid for 2,000 years before the appearance of Euclid's Elements. His work provided a framework for rational exploration, which was very important for scientists and philosophers during the 19th century. No serious objections were raised against Sir Isaac Newton's notion of fluxions in calculus. Berkeley's argument, raised by other scientists, did not call for a rethink of the basic foundations of the field. The discovery of alternative geometric designs during the 19th century, consistent with the most intuitively apparent assumptions, proved that the concept of Euclidean geometry could not correspond with reality. The work of Georg Cantor, who made significant discoveries in set theory, also highlighted the need for a more rigorous foundation in mathematics. This was because, if not appropriately addressed, the results of paradoxical tests would lead to further confusion. 

In ancient mathematics, propositions that required proof were usually only presented one at a time. Since a mathematician would only prove one of the proofs required, the others would be left to the reader. Later editors, such as Theon, would often provide their own proofs. During his time, the notations and mathematical ideas used in common currency limited the way he presented his work. For instance, he did not define an angle beyond two right angles. He also didn't use the product of multiple numbers. The treatment of number theory in geometric terms was possibly due to the Alexandrian system of numbers, which was incredibly awkward. 

The results of a given proposition are presented in a stylized manner, which is regarded as a common form of classical analysis. The first part of this process is the enunciation, which states the outcome in general terms. The setting-out follows, providing a figure and various geometrical objects with letters. The first step in presenting a proof is defining the figure, which then restates its importance in terms of the given figure. The next step is to extend the original figure to forward the proof, and then the "proof" follows. Finally, the "conclusion" is a statement that summarizes the conclusions that were drawn by the proof. The proofs results are not given out as to the method used to produce the result. Scholars have criticized the use of figures in the proofs of Euclid, arguing that he relied too much on the figures drawn rather than the underlying logic. However, based on the given configuration, the original proof of his proposition is generally regarded as valid.