Question: What was the result of identifying more and more chemically similar lanthanide elements?
About the article "Atomic number", section "The periodic table and a natural number for each element".

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The lanthanide or lanthanoid series of chemical elements comprises the fifteen metallic chemical elements with atomic numbers 57 through 71, from lanthanum through lutetium.
These fifteen lanthanide elements, along with the chemically similar elements scandium and yttrium, are often collectively known as the rare earth elements.
The informal chemical symbol Ln is used in general discussions of lanthanide chemistry to refer to any lanthanide.
All but one of the lanthanides are f-block elements, corresponding to the filling of the 4f electron shell; lutetium, a d-block element, is also generally considered to be a lanthanide due to its chemical similarities with the other fourteen.
All lanthanide elements form trivalent cations, Ln3+, whose chemistry is largely determined by the ionic radius, which decreases steadily from lanthanum to lutetium.
The lanthanide elements are the group of elements with atomic number increasing from 57 (lanthanum) to 71 (lutetium).
They are termed lanthanide because the lighter elements in the series are chemically similar to lanthanum.
Strictly speaking, both lanthanum and lutetium have been labeled as group 3 elements, because they both have a single valence electron in the d shell.
However, both elements are often included in any general discussion of the chemistry of the lanthanide elements.
In presentations of the periodic table, the lanthanides and the actinides are customarily shown as two additional rows below the main body of the table, with placeholders or else a selected single element of each series (either lanthanum and actinium, or lutetium and lawrencium) shown in a single cell of the main table, between barium and hafnium, and radium and rutherfordium, respectively.
This convention is entirely a matter of aesthetics and formatting practicality; a rarely used wide-formatted periodic table inserts the lanthanide and actinide series in their proper places, as parts of the table's sixth and seventh rows (periods). 
The lanthanide or lanthanoid (IUPAC nomenclature) series comprises the fifteen metallic chemical elements with atomic numbers 57 through 71, from lanthanum through lutetium.
These fifteen elements, along with the chemically similar elements scandium and yttrium, are often collectively known as the rare earth elements.
The informal chemical symbol Ln is used in general discussions of lanthanide chemistry.
All but one of the lanthanides are f-block elements, corresponding to the filling of the 4f electron shell; lutetium, a d-block element, is also generally considered to be a lanthanide due to its chemical similarities with the other fourteen.
All lanthanide elements form trivalent cations, Ln3+, whose chemistry is largely determined by the ionic radius, which decreases steadily from lanthanum to lutetium.
Between initial and final 6s2 electronic shells The lanthanide elements are the group of elements with atomic number increasing from 57 (lanthanum) to 71 (lutetium).
They are termed lanthanide because the lighter elements in the series are chemically similar to lanthanum.
Strictly speaking, both lanthanum and lutetium have been labeled as group 3 elements, because they both have a single valence electron in the d shell.
However, both elements are often included in any general discussion of the chemistry of the lanthanide elements.
In presentations of the periodic table, the lanthanides and the actinides are customarily shown as two additional rows below the main body of the table, with placeholders or else a selected single element of each series (either lanthanum or lutetium, and either actinium or lawrencium, respectively) shown in a single cell of the main table, between barium and hafnium, and radium and rutherfordium, respectively.
This convention is entirely a matter of aesthetics and formatting practicality; a rarely used wide-formatted periodic table inserts the lanthanide and actinide series in their proper places, as parts of the table's sixth and seventh rows (periods). 
Together with scandium and yttrium, the trivial name "rare earths" is sometimes used to describe all the lanthanides.
This name arises from the minerals from which they were isolated, which were uncommon oxide-type minerals.
However, the use of the name is deprecated by IUPAC, as the elements are neither rare in abundance nor "earths" (an obsolete term for water-insoluble strongly basic oxides of electropositive metals incapable of being smelted into metal using late 18th century technology) .
Cerium is the 26th most abundant element in the Earth's crust, neodymium is more abundant than gold and even thulium (the least common naturally occurring lanthanide) is more abundant than iodine, which is itself common enough for biology to have evolve critical usages for.
Despite their abundance, even the technical term "lanthanides" could be interpreted to reflect a sense of elusiveness on the part of these elements, as it comes from the Greek λανθανειν ("lanthanein"), "to lie hidden".
However, if not referring to their natural abundance, but rather to their property of "hiding" behind each other in minerals, this interpretation is in fact appropriate.
The etymology of the term must be sought in the first discovery of lanthanum, at that time a so-called new rare earth element "lying hidden" in a cerium mineral, and it is an irony that lanthanum was later identified as the first in an entire series of chemically similar elements and could give name to the whole series.
The term "lanthanide" was introduced by Victor Goldschmidt in 1925. 
The lanthanide contraction is responsible for the great geochemical divide that splits the lanthanides into light and heavy-lanthanide enriched minerals, the latter being almost inevitably associated with and dominated by yttrium.
This divide is reflected in the first two "rare earths" that were discovered: yttria (1794) and ceria (1803).
The geochemical divide has put more of the light lanthanides in the Earth's crust, but more of the heavy members in the Earth's mantle.
The result is that although large rich ore-bodies are found that are enriched in the light lanthanides, correspondingly large ore-bodies for the heavy members are few.
The principal ores are monazite and bastnäsite.
Monazite sands usually contain all the lanthanide elements, but the heavier elements are lacking in bastnäsite.
The lanthanides obey the Oddo-Harkins rule – odd-numbered elements are less abundant than their even-numbered neighbors.
Three of the lanthanide elements have radioactive isotopes with long half-lives (138La, 147Sm and 176Lu) that can be used to date minerals and rocks from Earth, the Moon and meteorites. 
Lanthanide triflates consist of a lanthanide metal ion and three triflate ions.
The lanthanides, or rare earth metals, are the elements from lanthanum to lutetium in the periodic table.
Triflate is a contraction of trifluoromethanesulfonate; its molecular formula is CF3SO3, and is commonly designated ‘OTf’.
Triflic acid is a ‘superacid’ so its conjugate base ions are very stable.
Lanthanide triflates are normally nonahydrates, mostly commonly depicted as Ln(OTf)3·(H2O)9; however, in the solid state and in aqueous solution, the waters are bound to the lanthanide and the triflates are counteranions, so more accurately lanthanide triflate nonahydrate is written as [Ln(H2O)9](OTf)3.
Anhydrous lanthanide triflates, Ln(OTf)3, are also easily obtained as described below.
The metal triflate complex is strongly electrophilic, thus acts as a strong Lewis acid.