In 1829, a German chemist by the name of J. W. Dobereiner noticed that certain groups of three elements had similar properties. He called these groups triads and published a system of classification based on them. For example, chlorine, bromine and iodine formed a triad, based on the fact that the atomic weight of bromine (79.904) was close to the average of the atomic weights of chlorine (35.453) and iodine (126.904). Unfortunately for Dobereiner and his scientific legacy, not all of the elements could be grouped into triads, so his efforts failed. Another classification system unsuccessfully attempted to group the elements into octaves, like musical notes.
In 1869, Russian chemist Dmitry Mendeleyev published the first periodic table of elements, writing the chemical properties and masses of each element on cards. He arranged the cards according to increasing atomic mass and found that elements of similar properties appeared at regular intervals. But he took some liberties with his table. In some cases, he violated his order of increasing atomic masses to keep elements with similar properties together. For example, he placed tellurium (atomic weight 128) before iodine (atomic weight 127), so that iodine could be grouped with chlorine, bromine and fluorine, all of which have properties similar to iodine. He also reasoned that if elements had to be reversed to preserve the periodic pattern, then the atomic mass values must be wrong. Lastly, he left gaps in his table for elements that he reasoned should exist, but hadn't been discovered.
Mendeleyev's periodic table predicted three elements of atomic weights 45, 68 and 70. He was proven right when these elements were later discovered and identified as scandium, gallium and germanium, respectively. The atomic weights listed in modern periodic tables are slightly different than those in Mendeleyev's time because methods for measuring atomic weights were improved during the 20th century. These discoveries demonstrated the usefulness of Mendeleyev's approach, even if it wasn't without problems. Explanations would have to wait until the early 20th century, when the structure of the atom began to be revealed.
In 1911, English chemist Henry Moseley studied the frequencies of X-rays given off by various elements when high-energy electrons bombarded each. The X-rays each element emitted had a unique frequency that increased with increasing atomic mass. Moseley arranged the elements in order of increasing frequency and assigned each one a number called the atomic number (Z). He realized that the atomic number was equal to the number of protons or electrons. When the elements were arranged by increasing atomic number, the periodic pattern was observed without having to switch some elements (as Mendeleyev did), and "holes" in the periodic table led to the discovery of new elements. Moseley's discovery was summarized as the periodic law: When elements are arranged in order of increasing atomic number, there's a periodic pattern in their chemical and physical properties. That law led to the modern periodic table.