Gemara

Suggestion: A Kal va'Chomer should permit tithing new animals on old, or vice-versa!

Goats and sheep are Kilayim with each other, yet one may tithe from one on the other. New and old animals are not Kilayim with each other. All the more so, we should be able to tithe from one on the other!

Rejection: "Aser Ta'aser" refers to two Ma'aseros and equates them, i.e. of grain and animals;

Just like one may not tithe one year's grain on another year's, one may not tithe one year's animals on another year's.

Rosh Hashanah 2a (Mishnah): Elul 1 is Rosh Hashanah for Ma'aser Behemah;

R. Elazar and R. Shimon say, it is Tishrei 1.

7a (Beraisa): Nisan 1 is Rosh Hashanah for Terumas (taking some of the) Shekalim (to bring Korbanos Tzibur from the new Shekalim).

7b: The Tana of our Mishnah omitted this. It was taught that if one brought from the old (Shekalim), he was Yotzei, so it is not clearcut.

Rishonim

Tosfos (Bechoros 33a DH u'Beis): Here, a Kal va'Chomer overrides a Hekesh. We use the Hekesh to teach other laws. Elsewhere, we say that a Kal va'Chomer overrides a Gezeirah Shavah. On 53b, the Hekesh overrides the Kal va'Chomer. That is because if we would follow the Kal va'Chomer, the Hekesh would not teach anything. Even though according to R. Yosi b'Rebbi Yehudah the Hekesh is needed to teach that Ma'aser is taken through estimation and mere intent, and we need it to exempt for Kodshei Nochrim, the Hekesh primarily teaches "Shanah Shanah" (that both are tithed from the same year's yield. This is based on Tzon Kodoshim's text.)

Rambam (Hilchos Bechoros 7:5): We may not tithe one year's animals on another year's, just like we do not tithe old produce on new, or vice-versa. It seems to me that if one tithed from one year on another it is Ma'aser, due to the stringency of Kodshim. The Torah did not explicitly say that Ma'aser must be Shanah Shanah.

Lechem Mishneh: This is difficult, for we learn from a Hekesh to Ma'aser of grain. Just like one cannot tithe grain from one year on another, the same applies to animals! A Hekesh is as if it is written explicitly. We learn (that we do not tithe) an orphaned animal, Mechusar Zman (before eight days, when it can be offered) and Kil'ayim from a Gezeirah Shavah from Kodshim. The Rambam did not say that if one tithed them, we are stringent to say that it is Ma'aser. We hold that a Hekesh is stronger than a Gezeirah Shavah, like it says in Halichos Olam. If so, why did the Rambam say that there he is totally exempt, and here he calls it Ma'aser due to the stringency of Kodshim? I answer that there is a Kal va'Chomer to tithe from new on old, like the Gemara says. Tosfos answered why here the Hekesh overrides the Kal va'Chomer. The Rambam said that the Torah did not explicitly say that Ma'aser must be Shanah Shanah, because there is a Kal va'Chomer against the Hekesh, and it is possible to fulfill both of them.

Turei Even (Rosh Hashanah 2a, Avnei Milu'im DH b'Echad b'Elul): Tishrei 1 is Rosh Hashanah. This means that we may not tithe those born before Tishrei 1 on those born after, or vice-versa. The Rambam says that if one did so, it is Ma'aser, due to the stringency of Kodshim, since the Torah did not explicitly say that Ma'aser must be Shanah Shanah. This is difficult, for every Rosh Hashanah mentioned in our Mishnah is Me'akev b'Di'eved! The Gemara said that our Tana omitted Terumas Shekalim, because if one brought from the old, he was Yotzei, so it is not clearcut. If you will say that b'Di'eved, Ma'aser Behemah from one year on another takes effect, the Tana should have omitted also Ma'aser Behemah! However, I explain 'since if one brought from the old Shekalim, he was Yotzei, so it is not clearcut' does not explain why the Tana did not teach Rosh Hashanah for Terumas Shekalim. Surely, we teach a Rosh Hashanah even if it is only l'Chatchilah. Rather, it means that if the new Shekalim did not come in time, we bring from the old Shekalim l'Chatchilah, therefore it is not always Rosh Hashanah, so it was not taught. Rosh Hashanah for Ma'aser Behemah was taught, because even if you will say that if one tithed from one year on another it is Ma'aser, this is never l'Chatchilah.

Or Some'ach: The Rambam says that if one took an animal from one year be Ma'aser on animals of another year, it is Ma'aser, but five from this year and five from another year do not join. If one tithed, it is not Ma'aser. Regarding this the Mishnah taught Rosh Hashanah for Ma'aser Behemah. The Mishnah omitted Rosh Hashanah for joining Shekalim to take Terumah from them, since this is only l'Chatchilah. The next Halachah teaches that (regarding Ma'aser Behemah, it is even b'Di'eved). We need not say like the Turei Even.

Sefas Emes (Rosh Hashanah 2a DH b'Rashi): The Tana would not omit teaching a day that is Rosh Hashanah, even if it is only for a matter that is not Me'akev. However, if a day is Rosh Hashanah for matters that are Me'akev, we do not list also matters that are not Me'akev. This answer the Turei Even's question.

Rambam (6): Everything born from Tishrei 1 until Elul 29 joins, and we tithe them on each other. If five were born on Elul 29 and five were born on Tishrei 1, they do not join.

Or Some'ach: In the Yerushalmi (Shekalim 3:1, Daf 8a), R. Chuna explains that R. Meir says that Elul 1 is Rosh Hashanah for Ma'aser Behemah because he holds that their year begins from Nisan, and the gestation of small animals in five months, so they give birth before Elul 1. However, this is difficult, since the months of pregnancy need not be complete for Tahor animals. If an animal become pregnant after Nisan 1, and gave birth before Elul 1, he will tithe from one year on another year! This is not difficult for the Rambam, who holds that Ma'aser takes effect in such a case. R. Meir fixed Elul 1 for Rosh Hashanah, since usually the pregnancy began before Nisan. This explains why Rava (58a-b) does not decree lest one take from the wrong animals, since even if he tithes from one year on another year, it is Ma'aser. Several Tana'im expound other verses to teach that one must tithe from the same year, except for R. Akiva, who expounds Aser Ta'aser. We can say that the Torah speaks the way people do (and we do not expound the repetition)! Therefore, it is not considered explicit.

Tosfos and Rosh (Kesuvos 50a DH Al, and 4:15): It was enacted not to squander more than a fifth of one's income (for Tzedakah), lest he need to take from others. The Yerushalmi says that the first time he may give a fifth, and afterwards a fifth of his income every year.

Poskim

Shulchan Aruch (YD 331:57): One may not tithe this year's Peros on last year's, or vice-versa. If he did so, it is not Terumah, for it says Shanah Shanah. One may not tithe one year's animals on another year's.

Question: Regarding Ma'aser Kesafim (tithing income), if one gained money in some business dealings and lost in others, may he deduct the losses from the gains? Or, must he tithe all the gains without deducting losses?

Answer (Noda bi'Yehudah 2 YD 198): There is no source in the Gemara for this. Logically, we do not join profits and losses. It is proper to tithe what Hash-m gives to him, even though there were also losses. This is like Ma'aser of grain and animals. If one has two fields, and one was blasted and perished, and the other was blessed, is he exempt from tithing the latter due to the loss to the former?! If one has two flocks, and all the animals in one flock died, does this exempt tithing the other?! However, Sha'ar Efrayim (84) compares it to a business venture, and concluded that if he already calculated the loss, he cannot deduct it from what he will profit afterwards, but if he did not calculate it yet, he may. This seems unreasonable.

Noda bi'Yehudah: It seems that if the profit and loss were in the same year, one may deduct the loss from the profits, but not if they were in different years. We learn Ma'aser Kesafim from "Aser Ta'aser Es Kol Tevu'as Zar'echa." "Kol" include business and Ribis. I say that also Shanah Shanah applies to Ma'aser Kesafim, to teach that each year is calculated by itself. For Ma'aser of grain and Ma'aser Behemah, it teaches that we do not tithe old on new (or vice-versa), i.e. from one year on another year. Old and new does not apply to money, therefore it teaches about deducting losses from profits. I bring a proof from Tosfos and the Rosh. They should have said 'the first time he may give a fifth, and afterwards a fifth of every profit.' Why did they say '...and afterwards a fifth of his income every year'? This teaches that every year is one matter. At the end of the year he calculates. We learn from Shanah Shanah. However, the Yerushalmi is not like Tosfos and the Rosh cite it. It says that there is a Shi'ur for Chesed with one's money. It was enacted to separate a fifth of one's property. It asked that in five years, one will not have anything left! It answered that first he separates from the principal, and afterwards from the earnings. It did not mention each year! However, also the Yerushalmi proves that one calculates every year. It suggested that after five years, one will not have anything left. Why didn't it say that this will happen after five days or five weeks? The Yerushalmi expounds Aser Ta'aser to teach about two tithes, which is a fifth in all.

Noda bi'Yehudah: Do not say that Tosfos and the Rosh mentioned a year regarding Bal Te'acher (the Isur to delay offering Korbanos and other obligations). One is liable immediately for (delaying) Tzedakah. Also, it seems that Bal Te'acher applies to Bechor and Kodshim, but not to Ma'aseros, Erchin and similar matters.