Halakha Ulangan 16:1: Mishneh Torah, Sanctification, Sefer HaChinukh & The Sabbath Epistle

The months of the year are lunar months, as it is stated (Numbers 28:14), "the monthly burnt-offering for each new moon"; and it is [also] stated (Exodus 12:2), "This month shall be for you the beginning of the months." Thus did the Sages say: The Holy One, blessed be He, showed Moshe a likeness of the moon in a prophetic vision and said to him, "See [when] it is like this, and sanctify it (as the beginning of the month)." But the years that we calculate are solar years, as it is stated (Deuteronomy 16:1), "Guard the month of the Spring."

The commandment of sanctifying the new month: To sanctify months and intercalate [months into] years in the court [whose members are] great in wisdom and ordained in the Land of Israel, and to establish the year’s holidays according to that sanctification; as it is stated (Exodus 12:2), “This month shall be to you the first of months.” This means, when you see the renewal of the moon, establish for yourselves the new month — or even if you do not see it, since it is appropriate for [the moon] to appear according to the accepted calculation. Likewise, this commandment includes the commandment of intercalation, since the basis of the commandment to sanctify the month is for Israel to observe God’s holidays at their appointed times. And the commandment of intercalation has the same basis. However, beyond this verse are more Torah passages concerning the commandment of intercalation, and that is what is written (Exodus 13:10), “You shall observe this commandment in its proper time”; and likewise, “Observe the month of Spring” (Deuteronomy 16:1).

A leap year is a year in which another month has been added. And we only ever add an Adar and make two Adars, an Adar Rishon (first Adar) and an Adar Sheini (Second Adar). And why do we add this month? Because of the spring time, so that the Passover will fall at that time, as it is said (Deuteronomy 16:1) "Guard the month of the spring", that this month should be in the spring. And if it wasn't for the addition of this month, Passover would sometimes come in the days of the sun (the summer) and sometimes in the days of the rain (the winter).