Lavoisier's list, called the Traité Élémentaire de Chimie, contains 33 elements, sorted into metals and non-metals. It was one of the first attempts to make a systematic method of sorting elements. However, it was not completely correct, including light and caloric as elements (it is now known that light is not made of atoms and caloric is, in fact, not real).
Döbereiner notices that elements in Lavoisier's list can be sorted into groups of three elements. All of these elements share similar properties, and the mass of the second element is almost exactly the average of the mass of the first and second elements.
Today, it is known that the "triads" he found were elements in the same group on the periodic table.
Alexandre-Emile Béguyer de Chancourtois makes an early periodic table, where the elements are arranged in a spiral. He noticed that elements with similar properties - such as the ones in Döbereiner's triads.
When classifying the 56 elements that had been discovered at the time, he noticed that properties seemed to reoccur in multiples of eight, and sorted the elements into different groups based upon this.
Mendeleev makes a presentation to the Russian Chemical Society called "The Dependence Between the Properties of the Atomic Weights of the Elements" where he presents his periodic table, with the elements arranged in octaves. It is then published in an obscure Russian journal, but is soon published again in the German Zeitschrift für Chemie.
Mendeleev also predicted some elements based upon gaps in his periodic table - these elements were discovered and called germanium, gallium and scandium.
After discovering a way to directly measure how many protons an elements has, Mosely reorders the table based on atomic number rather than mass (the same ordering is used in modern tables today). For example, argon was placed before potassium, which made the table more correct.
He also predicted the elements Technetium and Promethium, as there were new gaps in the table made after it was reordered.