Preface

About Conics and Synthetic Geometry

After half a century, I have fulfilled my dream to bring to light the study of conics. I revisited my high school notebooks from when I was 17 and preparing to be an architect. I was passionate about geometry, about precise constructions with ruler and compass. I started with projective geometry, then descriptive geometry, with complicated planes and sections. I arrived at the conics and went deeper, marveling at their miracle.

I built them one by one from shapes precisely cut from transparent plastic, then polished them until I reached the perfect form, drawn with a ruler and compass.

I found them again in Țițeica's geometry collection, which I went through twice. I learned their properties, wrote them nicely in a graph paper notebook, and drew them with explanations.

Then I transferred them to a large A4 format, with precise drawings made with a black pen, without corrections. This is how I learned them, carried them with me across countries and seas, and every time I leafed through them, I sighed and regretted leaving them scattered in piles that had yellowed with time.

I resumed them in Oregon last year, 2024, and now in Florida, I have finished them.

I wanted to bring new knowledge that is not taught in schools. While the parabola is studied in algebra, its equation is learned in the 8th grade, whether it holds water or not, its roots, real or imaginary.

These are well-known to all diligent students who have completed 10 grades. Similarly, in high school mechanics, I learned about projectile trajectory and free fall as applications of Newton's and Galileo's laws.

In analytical geometry, I studied all conics in turn, with equations and tangents. I bring novelties that are not taught in schools, perhaps only in mathematics or by architects in their first years of study.

The beauty and miracle of simple and precise constructions with ruler and compass, geometric loci, radical axes, fascinated me, and I went further and further.

Now I have found GeoGebra, which allows for precision with 6 decimal places, and you can create anything you want in geometry and graphs, and even more, solids and complicated intersections. This is how I wrote The History of Logarithms and Regular Polyhedra, where I inscribed the 5 Platonic solids in a sphere.

Likewise, I delighted in the pentagon and rediscovered it in the logarithmic spiral, which led me to the complex plane and Euler's formula. As a novelty, after studying the geometry of conics in detail, I penetrated them more deeply and found their connection with the Bible – the divine Word, descended from heaven.

Thus, I connected the hyperbola to its shadow and universe, the ellipse, I found it in "what is missing from the ellipse," then the parabola, I associated it with the Lord's parables from the New Testament.

I did not touch the circle; it is studied in schools, and through affinity, I moved on to the ellipse and the calculation of areas.

Of course, the study can continue at higher levels, but I stopped at the simplicity of simple and precise constructions that the Greeks used two millennia ago.

It is a fragment of my history, when I spent summer vacations with complicated geometries, when my colleagues rejoiced that they had escaped school and exams.

I was a daring person, passionate about mathematics, and I strived to reach higher and higher. I did not use the geometry of conics, but I remained with pleasant memories, and now I see them gathered in a modest little book.

Pavy Beloiu, Florida - May 19, 2025